Radar detection of migrating targets using an interference correlation matrix

ABSTRACT

A computer-implemented method is provided for detecting a target amidst clutter by a radar system able to transmit an electromagnetic signal, receive first and second echoes respectively from the target and the clutter, and process the echoes. The method includes determining signal convolution matrix for the target and a target return phase, clutter amplitude by spatial correlation matrix of clutter, clutter correlation matrix, receive noise power; querying whether the clutter moves as a motion condition if satisfied and as a stationary condition otherwise; calculating signal convolution matrix and target return phase from the signal convolution matrix and the target return phase for target motion; querying whether the target has range migration as a migration condition if satisfied and as a non-migration condition otherwise; and forming a target detector for the radar. The motion condition further includes calculating signal convolution matrix from clutter motion, clutter range migration matrix from the clutter motion, and interference correlation matrix. The stationary condition further includes calculating the interference correlation. The migration condition further includes calculating range migration matrix from the target motion.

STATEMENT OF GOVERNMENT INTEREST

The invention described was made in the performance of official dutiesby one or more employees of the Department of the Navy, and thus, theinvention herein may be manufactured, used or licensed by or for theGovernment of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefor.

BACKGROUND

The invention relates generally to radar detection of targets. Inparticular, the invention relates to detecting a target that movesrelative to the radar, thereby altering its range.

SUMMARY

Conventional radar detection techniques yield disadvantages addressed byvarious exemplary embodiments of the present invention. In particular,various exemplary embodiments provide computer-implemented method isprovided for detecting a target amidst clutter by a radar system able totransmit an electromagnetic signal, receive first and second echoesrespectively from the target and the clutter, and process the echoes.The method includes determining signal convolution matrix for the targetand a target return phase, clutter amplitude by spatial correlationmatrix of clutter, clutter correlation matrix, receive noise power;querying whether the clutter moves as a motion condition if satisfiedand as a stationary condition otherwise; calculating signal convolutionmatrix and target return phase from the signal convolution matrix andthe target return phase for target motion; querying whether the targethas range migration as a migration condition if satisfied and as anon-migration condition otherwise; and forming a target detector for theradar. The motion condition further includes calculating signalconvolution matrix from clutter motion, clutter range migration matrixfrom the clutter motion, and interference correlation matrix. Thestationary condition further includes calculating the interferencecorrelation. The migration condition further includes calculating rangemigration matrix from the target motion.

BRIEF DESCRIPTION OF THE DRAWINGS

These and various other features and aspects of various exemplaryembodiments will be readily understood with reference to the followingdetailed description taken in conjunction with the accompanyingdrawings, in which like or similar numbers are used throughout, and inwhich:

FIG. 1 is an elevation diagram view of radar pulse returns from a slowtarget;

FIG. 2 is an elevation diagram view of radar pulse returns from a mediumspeed target;

FIG. 3 is a first elevation diagram view of radar pulse returns from afast target;

FIG. 4 is a second elevation diagram view of radar pulse returns from afast target;

FIG. 5 is a schematic diagram view of radar hardware;

FIGS. 6A, 6B and 6C are plan diagram views of ground reference pointsfor radar;

FIG. 7 is a graphical view comparing an optimum detector againstCorrelator+MTD for normalized Doppler frequency;

FIG. 8 is a graphical view of comparing the optimum detector for a slowtarget;

FIG. 9 is a graphical view of comparing the optimum detector for a fasttarget;

FIG. 10 is a graphical view of optimum detector performance with seastate for a slow target and identical pulses;

FIG. 11 is a graphical view of optimum detector performance with noisefor a slow target and identical pulses;

FIG. 12 is a graphical view of optimum detector performance with seastate for a fast target and identical pulses;

FIG. 13 is a graphical view of optimum detector performance with noisefor a fast target and identical pulses;

FIG. 14 is a graphical view of optimum detector performance with noisefor non-identical versus identical pulses;

FIG. 15 is a tabular view of radar parameters;

FIGS. 16A, 16B, 16C, 16D, 16E and 16F are tabular views of alexicographical list of symbols and definitions; and

FIGS. 17A, 17B and 17C are a flowchart view of radar signal processoperations for detecting a target.

DETAILED DESCRIPTION

In the following detailed description of exemplary embodiments of theinvention, reference is made to the accompanying drawings that form apart hereof, and in which is shown by way of illustration specificexemplary embodiments in which the invention may be practiced. Theseembodiments are described in sufficient detail to enable those skilledin the art to practice the invention. Other embodiments may be utilized,and logical, mechanical, and other changes may be made without departingfrom the spirit or scope of the present invention. The followingdetailed description is, therefore, not to be taken in a limiting sense,and the scope of the present invention is defined only by the appendedclaims.

In accordance with a presently preferred embodiment of the presentinvention, the components, process steps, and/or data structures may beimplemented using various types of operating systems, computingplatforms, computer programs, and/or general purpose machines. Inaddition, artisans of ordinary skill will readily recognize that devicesof a less general purpose nature, such as hardwired devices, may also beused without departing from the scope and spirit of the inventiveconcepts disclosed herewith. General purpose machines include devicesthat execute instruction code. A hardwired device may constitute anapplication specific integrated circuit (ASIC), a field programmablegate array (FPGA), digital signal processor (DSP) or other relatedcomponent. The disclosure generally employs quantity units with thefollowing abbreviations: signal strength in decibels (dB), energy injoules (J), and frequencies in hertz (Hz).

Section I— Clutter Cancellation and Correlation: There is currently nomethod to enable a radar to suppress clutter under the situation thattargets (or clutter) move across range cells over the time of a coherentprocessing interval (CPI). The reason for this is that compensating forrange walk (or range migration) decorrelates the clutter. Cluttercancellation depends on the correlation properties of clutter. Hence,decorrelating the clutter destroys the radar's ability to cancelclutter. This problem can be visualized with some simple illustrations.

FIG. 1 shows an elevation diagram view 100 of target and clutter returnsfor low speed low acceleration targets 110 to be distinguished fromclutter 120 (that is stationary) by a radar platform 130, whichtransmits pulses having waveform profiles for Fast Time 140 and SlowTime 150. The radar 130 emits a first transmitted waveform 160 andreceives a first target reflection waveform 170 and a first clutterreflection waveform 175. Subsequently, the radar 130 emits a secondtransmitted waveform 180 and receives a second target waveform 190 and asecond clutter waveform 195. The shapes of the returned waveforms 175and 170 are nearly identical between clutter 120 and the target 110.Additionally, the subsequent pulses 190 from the target 110 occur in thesame range cells as the first received pulse 170. However, there is aDoppler difference in the target return 190 and the clutter return 195that enables the radar 130 to detect the target 110 and reject theclutter 120.

FIG. 2 shows an elevation diagram view 200 for a target 210 havingmedium speed and medium acceleration for detection by the radar 130against the stationary clutter 120 with waveform profiles analogous toview 100. The radar 130 emits a first transmitted waveform 220 andreceives a first target reflected waveform 230 and a first clutterreflected waveform 240. Subsequently the radar 130 emits a secondtransmitted waveform 250 and receives a second target reflected waveform260 and a second clutter reflected waveform 270. The target 210 causesrange migration, but pulse distortion can be ignored. The shapes of thereturned waveforms 240 and 230 are still nearly identical betweenstationary clutter 120 and the target 210. However, subsequent pulses260 from the target 210 occur in different range cells than the firstreceived pulse 230, producing an offset. Also, there is an even largerDoppler for the target 210 that may change from pulse-to-pulse due toacceleration.

FIG. 3 shows an elevation diagram view 300 for a target 310 having highspeed and high acceleration for detection by the radar 130 against thestationary clutter 120 with waveform profiles analogous to view 100. Theradar 130 emits a first transmitted waveform 320 and receives a firsttarget reflected waveform 330 and a first clutter reflected waveform340. Subsequently the radar 130 emits a second transmitted waveform 350and receives a second target reflected waveform 360 and a second clutterreflected waveform 370. The target's motion causes range migrationanalogous to view 200, as well as pulse distortion. The shape of thereturned waveform 330 from the target 310 differs significantly fromreturned waveform 340 from the stationary clutter 120. Additionally,subsequent pulses 360 from the target 310 occur in different range cellsas the first received pulse 330, producing an offset. Also, there is aneven larger Doppler for the target 310 that may change frompulse-to-pulse due to acceleration.

FIG. 4 shows an elevation diagram view 400 for a target 410 having highspeed and high acceleration for detection by the radar 130 against thestationary clutter 120 with waveform profiles analogous to view 100. Theradar 130 emits a first transmitted waveform 420 and receives a firsttarget reflected waveform 430 and a first clutter reflected waveform440. Subsequently the radar 130 emits a second transmitted waveform 450and receives a second target reflected waveform 460 and a second clutterreflected waveform 470. As in view 300, the target return pulses 430 and460 have different (i.e., shorter fast time) waveforms than clutterreturn pulses 440 and 470. Also, the clutter 120 appears to shift basedon the offset of the reflected waveforms 440 and 470, thereby presentinga phantom position 480. View 400 shows range migration compensation andits effect on clutter 120. Conventional signal processing time alignsdata from each pulse to ensure that the target return pulses 430 and 460appear in the same range cell for each transmit pulse 420 and 450, andcan compensate for the change impulse shape. This permits the radar 130to integrate the target returns. Unfortunately, this causes the clutterreturns 440 and 470 to move range cells for each transmit pulse 420 and450. When this happens, the effect destroys the radars ability to cancelthe clutter 120 via Doppler processing.

One can begin with the case where the target motion over the course ofthe coherent processing interval (CPI) is small compared to the rangeresolution of the radar 130. This case is illustrated in view 100, whichillustrates fast time/distance 140 being represented by the horizontaldimension. The vertical dimension represents slow time 150 (i.e., timebetween pulses of the radar). View 100 shows the radar 130 transmittingtwo pulses 160 and 180. The radar's field of view includes an airplane,which is moving and thereby represents a target 110, and a building,which remains stationary and represents clutter 120. The target motionis slow enough that it appears not to change range from the time of onepulse transmission to the next transmission. View 200 illustrates thecase where the target 210 is flying fast enough that the radar 130observes it in different range cells from on pulse to the next. That isthe range change from the time of the first transmitted pulse to thesecond transmitted pulse is observable.

The Doppler processing integrates energy pulse-to-pulse. In other words,it sums energy for a given range cell across slow time. Therefore, inthe case of view 200, the target energy appears to leak between rangecells, thus it is lost. Therefore, if range migration is not compensatedthere is a signal loss on the target. View 300 illustrates the case oftargets 310 traveling so fast that in addition to range migration, thereflected pulse shape is also distorted. This is caused by theobservable motion over the course of a pulse width. In this disclosure,the narrow band assumption is used. There are techniques to compensatefor range migration and pulse distortion. However, performing thiscompensation on the received data decorrelates radar clutter. This isillustrated in view 400. As shown, the target returns are aligned, andconsequently the clutter returns are misaligned causing seriousdegradation in clutter cancellation.

FIG. 5 shows a schematic diagram view 500 of hardware for the radar 130to detect a generic target 510. A transmitter 520 includes a signaloscillator 530 and a transmit antenna 535. The transmitter 520 emits atransmit pulse signal 540 that the target 510 reflects as a reflectionsignal 545. The pulse signal 540 can be repeated at intervalscorresponding to a pulse repetition frequency (PRF). A receiver 550receives the reflection signal 545 by a receive antenna 555. A low noiseamplifier (LNA) 560 divides the received signal into real and imaginarycomponents to corresponding pairs of mixers 565, low pass filters (LPF)570, switches 575 and analog-to-digital (A/D) converters 580 to adetector 590 into their respective real (Re) and imaginary (Im) signals.

The radar hardware is diagramed in view 500. The radar transmitter 520generates and amplifies the transmit signal 540 which is radiatedthrough the antenna 535. The reflection signal 545 then propagates tothe target 510 and then is reflected from the target 510 and is receivedby the antenna 555. The LNA 560 amplifies the received signal. Followingamplification, this signal is synchronously down converted to baseband;producing the complex in-phase (real) and quadrature (imaginary) signalsthat are available to be processed in order to detect the target 510.Following the down conversion, the analog signals are digitized at theNyquist rate. The output of the A/D converters 580 enables a detectiondecision to be made. The transmitter 520 transmits and receives multiplecoherent pulses 540 in order to make a detection decision. The receiver550 is gated on (i.e., switches 575 closed) during the listen time foreach pulse. The radar 130 maintains timing to align in time and phasethe fast and slow time data. Note that fast time data are the receiveroutput due to the echoes produced by one pulse. Slow time 150 indexesthe subsequent periods of fast time 140.

Section II—Slow Moving Target: The first detector 590 developed will befor a slow moving target (i.e., range migration can be ignored). Thisdetector 590 is optimal in that it maximizes the probability ofdetection for a specified probability of false alarm subject to theassumptions invoked in the text. The derivation jointly processes thefast time and slow time information to improve detectability. The inputof this detector 590 is the complex baseband data as shown in view 500.

Subsection (a) Single Pulse: First presented is the case for a constantvelocity target with a range rate and CPI time that satisfies:

$\begin{matrix}{{{{\overset{.}{R}(t)}{MT}_{i}} ⪡ \frac{{cT}_{s}}{2}},} & (1)\end{matrix}$where R(t) is the target range as a function of time {dot over(R)}(t)=dR(t)/dt is the target range rate as a function of time, M isthe number pulses, T_(i) is the time between pulses, T_(s) is the sampletime, and c is the speed of light. Further, this analysis is at complexbaseband. The transmitter 520 of the radar 130 transmits signal s(t) 540in the direction of the target 510. The response of the receiver 550 tothe reflected signal 545 is:

$\begin{matrix}{{{y_{i}(t)} = {{s(t)}*\alpha\;{h_{i}\left( {t - \frac{2{R\left( t_{0} \right)}}{c}} \right)}}},} & (2)\end{matrix}$where α is the complex amplitude of the target 510, h_(t)(t) is theimpulse response of the target 510, and t₀ is the start of the CPI. Forpoint targets h_(t)(t) is the Dirac delta function δ(t). Because theradar's receiver 550 produces data sampled by an A/D converter 580, theconvolution in eqn. (2) can be approximated by a discrete timeconvolution as follows:

$\begin{matrix}{{{y_{i}\left( {kT}_{s} \right)} = {\alpha{\sum\limits_{i = 0}^{N - 1}\;{{s\left( {iT}_{s} \right)}{h_{i}\left( {{\left( {k + 1 - i} \right)T_{s}} - \frac{2{R\left( t_{0} \right)}}{c}} \right)}}}}},} & (3)\end{matrix}$where T_(s) is the sample time interval and N is the total number oftime samples that contain the transmitted pulse 540.

The operation in eqn. (3) can be written in matrix notation as:y _(t) =α{tilde over (S)}h _(t)  (4)where y_(t) is a time-sampled receive signal column vector of size(P+N−1) and whose elements are the discrete time samples ofy_(t)(iT_(s)) and h_(t) is a target impulse column vector of size(P+2(N−1)) and whose elements are the time samples of h_(t)(iT_(s)−2R/c)in reverse range order and:

$\begin{matrix}{{\overset{\sim}{S} = \begin{bmatrix}0 & \ldots & 0 & s^{t} \\0 & \ldots & s^{t} & 0 \\\; & \ddots & \; & \; \\s^{t} & 0 & \ldots & 0\end{bmatrix}},} & (5)\end{matrix}$which is a rectangle matrix with a size of (P+N−1)×(P+2(N−1)). Thematrix {tilde over (S)} is sometimes called a convolution matrix. Notethat NT, is the pulse width, (P+N−1) is the length of the received data,and (P+2(N−1)) is the number of range cells that influence theinter-pulse period (IPP). The received data's length constitutes thenumber of samples received during the IPP. Note for a point target, allelements of target impulse vector h_(t) are zero except the element thatcorresponds to the range cell that the target 510 is within. Thiselement will be unity.

The target model in eqn. (4) is adequate for stationary targets ortargets 110 whose Doppler can be ignored. The target Doppler manifestsitself in the receive data as a phase change from sample to sample. Thiscan be accounted for by modifying the target model as follows:y _(t) =α{tilde over (S)} _(d)(m)h _(t),  (6)where m is the pulse number that becomes relevant in Subsection (b),

$\begin{matrix}{{{{\overset{\sim}{S}}_{d}(m)} = \begin{bmatrix}0 & \cdots & s_{1} & {s_{2}e^{j\;{\phi{({m,1})}}}} & \cdots & {s_{N}e^{j\;{\phi{({m,{N - 1}})}}}} \\\; & \; & \; & \vdots & \; & \; \\s_{1} & {s_{2}e^{j\;{\phi{({m,1})}}}} & \cdots & {s_{N}e^{j\;{\phi{({m,{N - 1}})}}}} & \cdots & 0\end{bmatrix}},} & (7)\end{matrix}$where s₁ are the elements of sample vector s of the radar's transmitpulse, and target phase angle is:

$\begin{matrix}{{{\phi\left( {m,n} \right)} = \frac{4{\pi\left( {{mT}_{i} + {nT}_{s} - {R\left( {mT}_{i} \right)}} \right.}}{\lambda}},} & (8)\end{matrix}$where λ is the wavelength. In eqn. (6), the target's position andimpulse response is represented in h_(t) and the target's Doppler isrepresented in {tilde over (S)}_(d)(m). Note that if the target 510 isconstant velocity, then phase angle ϕ will only be a function of n. In asimilar manner the receiver response to the clutter 120 is:y _(c)(t))=s(t)*c(t),  (9)where c(t) is the fast time complex random process due to clutterback-scatter.

Similarly, the sampled response to clutter 120 can be written as:y _(c) ×{tilde over (S)}c,  (10)where the clutter vector c is a column vector of size (P+2(N−1)) whoseelements are the complex clutter impulse response in reverse range(i.e., fast time) order. The receiver's response to noise can be writtenas:y _(n) =n,  (11)where y_(n) and n are column vectors of length (P+N−1). The noise isassumed to be zero mean additive white Gaussian noise (AWGN). Thecombined receiver response then is:y=y _(t) +y _(c) +y _(n) =α{tilde over (S)} _(d) h _(t) +{tilde over(S)}c+n.  (12)The interference response of the receiver is:y _(l) =y _(c) +y _(n) ={tilde over (S)}c+n.  (13)The correlation matrix for the interference process can be determinedas:R _(l) =E{y _(l) y _(l) ^(H) }=E{({tilde over (S)}c+n)({tilde over(S)}c+n)^(H)},  (14)where E{ } denotes an expectancy function, and the superscript Hindicates conjugate transpose.

By assumption, both clutter vector c and noise vector n are zero meanand independent processes that gives:E{({tilde over (S)}cn ^(H))}=E{nc ^(H) {tilde over (S)} ^(H)}=0.  (15)Additionally, vector n is AWGN, which gives:E{(nn ^(H))}=σ_(n) ² I,  (16)where σ_(n) ² of is the noise variance (i.e., power) and I is a(P+N−1)×(P+N−1) identity matrix. Next, one can assume that the cluttervoltages are uncorrelated cell to cell because the clutter voltage fromeach cell has a random initial phase. This phase is uniformlydistributed zero to 2π and will be inter-cell independent. This givesspatial correlation matrix for the clutter 120 as:

$\begin{matrix}{{R_{c} = {{E\left\{ {cc}^{H} \right\}} = \begin{bmatrix}\sigma_{1}^{2} & 0 & \cdots & 0 \\0 & \sigma_{2}^{2} & \cdots & 0 \\\; & \; & \vdots & \; \\0 & 0 & \cdots & \sigma_{P + {2{({N - 1})}}}^{2}\end{bmatrix}}},} & (17)\end{matrix}$where σ_(i) ² is the clutter variance (power) at range cell i. Havingeqn. (17) now enables one to write the interference correlation matrixas:R _(l) =R _(y) _(c) +R _(y) _(c) ={tilde over (S)}R _(c) {tilde over(S)} ^(H)σ_(n) ² I.  (18)Clutter seen by a moving radar is addressed in Section III Subsection(a).

One can now invoke the assumption that clutter 120 is a compoundGaussian process. The compound Gaussian model states that the cluttervoltage at any range cell i at time i is defined as:c _(i)(1)=σ_(i) g _(i)(t),  (19)where σ_(i) is a random variable that is equal to the square root of thevariance (power) at range cell i, and g_(i)(t) is a unity variancecomplex Gaussian process that accounts for the pulse-to-pulse varianceof the clutter complex amplitude. The correlation function of theclutter 120 is:R _(g) ^(i)(τ)=E{g _(i)(t+τ)g* _(i)(t)},  (20)which defines the clutter spectral characteristics. The autocorrelationof a random process and its power spectral density form a Fouriertransform pair. The process σ_(i) is often called the texture, andg_(i)(t) is called the speckle. This model accounts for the significantchanges of clutter amplitude from range cell to range cell as well asthe Doppler spectrum properties of the clutter 120. The clutter'sDoppler spectrum then is the Fourier transform of the time correlationfunction. If the clutter 120 is compound Gaussian, and the cluttervariance is known, then the resulting clutter distribution is Gaussian.In other words, if clutter variance σ_(i) ² is known (through onlinemeasurement and estimation or through clutter modeling) then theclutter's spatial correlation matrix R_(c) in eqn. (17) is known. Thisin turn means the interference process as perceived by the radarreceiver in y_(l) is a Gaussian random process whose correlation matrixis determined in eqn. (18). If interference process is Gaussian, thenthe Neyman-Pearson (N-P) detector 590 can be formed by:

$\begin{matrix}{{{{h_{i}^{H}{\overset{\sim}{S}}_{d}^{H}R_{l}^{- 1}y}}_{\underset{H_{0}}{<}}^{\overset{H_{1}}{>}}\eta},} & (21)\end{matrix}$where the value of detection threshold η is chosen to achieve thedesired probability of false alarm or P_(fa).

Subsection (b) Multiple Pulses: The receiver response to the target 510is first determined. Under the slow moving target assumption, the targetresponse is identical from pulse-to-pulse except for the phase changethat is imparted due to the targets motion from pulse-to-pulse. Thus thereceiver response to the target 510 can be represented as stackedvector:

$\begin{matrix}{{Y_{t} = {\alpha\begin{bmatrix}{u_{0}{{\overset{\sim}{S}}_{d}(0)}h_{i}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}h_{i}}\end{bmatrix}}},} & (22)\end{matrix}$where u_(i) accounts for the phase change from pulse-to-pulse and iscomputed as:

$\begin{matrix}{{u_{i} = {\exp\left( {j\; 4\pi\frac{R\left( {iT}_{i} \right)}{\lambda}} \right)}},} & (23)\end{matrix}$and λ is the wavelength and T_(i) is the pulse repetition interval(PRI). (Recall that j is the imaginary number √{square root over (−1)}.)The phase change is proportional to the range change as a function oftime and is generally ascribed to the Doppler effect. For the case of anon-accelerating targets (i.e, the signal convolution matrix {tilde over(S)}_(d) is independent of pulse number), the Kronecker product enableseqn. (22) to be written compactly as:Y _(t) =u {tilde over (S)} _(d) h _(t),  (24)where u is a column vector whose elements defined in eqn. (23). The sizeof phase change vector u is M×1, where M is the number of pulses.Therefore, the size of time-sampled receive data matrix Y, isM(N+P−1)×1.

Next the clutter pulse-to-pulse is determined. Using the compoundGaussian model the time correlation function is designated R_(R)(τ)defined in eqn. (17). If one assumes that the clutter 120 over the wholerange extent is the same type (e.g., sea clutter), then it is notunrealistic to expect that clutter correlation R_(g)(τ) is the same forevery range cell. This means that for every range cell i there is arandom draw of the random variable σ_(i) determining its variance(power) and a random draw of the stationary random process g(t) that iszero mean, unity variance complex Gaussian that is correlated in timeaccording to R_(g)(τ). Due to the random and independent initial phaseof the clutter voltage, R_(g)(τ) is uncorrelated from range cell torange cell as previously discussed. Also, one can assume that thecomplex Gaussian random processes from one range cell to the next areuncorrelated.

Under these assumptions, one can define c(t) as the column vector of theclutter amplitudes at slow time i (noting that c(t) is a complex randomprocess). One writes the stacked vector representing the response of thereceiver due to clutter 120 as:

$\begin{matrix}{Y_{c} = {\begin{bmatrix}{\overset{\sim}{S}{c(0)}} \\{\overset{\sim}{S}{c\left( {\left( {i - 1} \right)T_{i}} \right)}} \\\vdots \\{\overset{\sim}{S}{c\left( {\left( {M - 1} \right)T_{i}} \right)}}\end{bmatrix}.}} & (25)\end{matrix}$The clutter correlation matrix for the stacked vector Y_(c) isdetermined as:

$\begin{matrix}{R_{Y_{c}} = {{E\left\{ {Y_{c}Y_{c}^{H}} \right\}} = {E{\begin{Bmatrix}{\overset{\sim}{S}{c(0)}{c^{H}(0)}{\overset{\sim}{S}}^{H}} & \ldots & {\overset{\sim}{S}{c(0)}{c^{H}\left( {\left( {M - 1} \right)T_{i}} \right)}{\overset{\sim}{S}}^{H}} \\\; & \ddots & \; \\{\overset{\sim}{S}{c\left( {\left( {M - 1} \right)T_{i}} \right)}{c^{H}(0)}{\overset{\sim}{S}}^{H}} & \ldots & {\overset{\sim}{S}{c\left( {\left( {M - 1} \right)T_{i}} \right)}{c^{H}\left( {\left( {M - 1} \right)T_{i}} \right)}{\overset{\sim}{S}}^{H}}\end{Bmatrix}.}}}} & (26)\end{matrix}$

This can be rewritten as:

$\begin{matrix}{R_{Y_{c}} = {\begin{Bmatrix}{\overset{\sim}{S}E\left\{ {{c(0)}{c^{H}(0)}} \right\}{\overset{\sim}{S}}^{H}} & \ldots & {\overset{\sim}{S}E\left\{ {{c(0)}{c^{H}\left( {\left( {M - 1} \right)T_{i}} \right)}} \right\}{\overset{\sim}{S}}^{H}} \\\; & \ddots & \; \\{\overset{\sim}{S}E\left\{ {{c\left( {\left( {M - 1} \right)T_{i}} \right)}{c^{H}(0)}} \right\}{\overset{\sim}{S}}^{H}} & \ldots & {\overset{\sim}{S}E\left\{ {{c\left( {\left( {M - 1} \right)T_{i}} \right)}{c^{H}\left( {\left( {M - 1} \right)T_{i}} \right)}} \right\}{\overset{\sim}{S}}^{H}}\end{Bmatrix}.}} & (27)\end{matrix}$Each expectation in eqn. (24) can be represented as:

$\begin{matrix}{{E\left\{ {{c\left( {jT}_{i} \right)}{c^{H}\left( {kT}_{i} \right)}} \right\}} = {\begin{bmatrix}{\sigma_{1}^{2}{R_{g}^{1}\left( {\left( {j - k} \right)T_{i}} \right)}} & \; & 0 \\\; & \ddots & \; \\0 & \; & {\sigma_{P + {2{({N - 1})}}}^{2}{R_{g}^{P + {2{({N - 1})}}}\left( {\left( {j - k} \right)T_{i}} \right)}}\end{bmatrix}.}} & (28)\end{matrix}$The result in eqn. (28) is obtained by applying eqn. (17) and invokingthe assumption that the clutter 120 is zero mean and uncorrelated cellto cell.

Under the assumption that the clutter 120 has the same Doppler spectrumin each cell, it can be further simplified to eqn. (24). In that casedefine the time correlation coefficient as:ρ_(j,k) =R _(g)((j−k)T _(i)).  (29)This enables eqn. (24) for the receive correlation matrix to be writtenas:

$\begin{matrix}{{R_{y_{c}} = {\begin{bmatrix}{\rho_{1,1}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}} & \; & {\rho_{1,M}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}} \\\; & ⋰ & \; \\{\rho_{M,1}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}} & \; & {\rho_{M,M}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}}\end{bmatrix} = {{M_{c} \otimes \overset{\sim}{S}}R_{c}{\overset{\sim}{S}}^{H}}}},} & (30)\end{matrix}$where the time correlation matrix for clutter is:

$\begin{matrix}{M_{c} = {\begin{bmatrix}\rho_{1,1} & \; & \rho_{1,M} \\\; & \ddots & \; \\\rho_{M,1} & \; & \rho_{M,M}\end{bmatrix}.}} & (31)\end{matrix}$Remember that spatial correlation matrix R_(c) (e.g., amplitude) isdiagonal because clutter 120 is uncorrelated from range cell to rangecell. Note also that time correlation matrix M_(c) is size M×M.

The complete interference correlation matrix can be written from eqn.(18) as:R _(l) =R _(y) _(c) +σ_(n) ² I,  (32)where R_(y) _(c) is defined in eqns. (30) and (31). Note that R_(l),R_(y) _(c) , and I are size M(P+N−1)×M(P+N−1). Under the assumption thatthe Doppler spectrum is the same for all range cells correlation matrixfor the interference process can be written as:R _(t) =M _(c) ß{tilde over (S)}R _(c) {tilde over (S)} ^(H)+σ_(n) ²I.  (33)With the signal model and interference model established the N-Pdetector 590 for the h_(t) can be written as:

$\begin{matrix}{{{{\begin{bmatrix}{u_{0}{{\overset{\sim}{S}}_{d}(0)}h_{i}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}h_{i}}\end{bmatrix}^{H}R_{I}^{- 1}Y}}\begin{matrix}\overset{H_{1}}{>} \\\underset{H_{0}}{<}\end{matrix}\eta},} & (34)\end{matrix}$where η is the detection threshold which is set to the specifiedprobability of false alarm. The scalars H₀ and H₁ correspond to thelower and upper values that determine whether the alarm threshold hasbeen satisfied.

Next the case of non-homogeneous clutter 120 is considered. Thissituation arises when there are clutter boundaries, such as the boundarybetween land and sea. This situation can be handled by defining thediagonal time correlation matrix ρ_(j,k) of size (P+2(N−1))×(P+2(N−1)).The diagonal elements are determined as:ρ_(j,k) =R _(g) ^(i)((j−k)T _(i)),  (35)where time correlation of clutter R_(g)(t) is defined in eqn. (20) and iindicates the range cell number. With this, eqn. (28) can be rewrittenas:E{c(jT _(i))c ^(H)(kT _(i))}==ρ_(j,k) R _(c).  (36)With eqn. (36), the interference correlation matrix can be rewritten as:

$\begin{matrix}{R_{I} = {\begin{bmatrix}{\overset{\sim}{S}\rho_{i,1}R_{c}{\overset{\sim}{S}}^{H}} & \; & {\overset{\sim}{S}\rho_{1,M}R_{c}{\overset{\sim}{S}}^{H}} \\\; & \ddots & \; \\{\overset{\sim}{S}\rho_{M,1}R_{c}{\overset{\sim}{S}}^{H}} & \; & {\overset{\sim}{S}\rho_{M,M}R_{c}{\overset{\sim}{S}}^{H}}\end{bmatrix} + {\sigma_{n}^{2}{I.}}}} & (37)\end{matrix}$

Subsection (c)—Fast Target: For this case, the target 310, 410 is movingso fast that its movement pulse-to-pulse relative to the sample intervalsuch that the rate inequality from eqn. (1) is not satisfied. That isthe target will move a significant fraction of a range cell from onepulse to the next. This requires some form of range migrationcompensation such that the target returns can be integrated up. Thetarget model first addresses the range migration issue frompulse-to-pulse. Then the effect of pulse distortion is caused by rangemigration within a pulse receive time. Finally, the Doppler effect isincluded with range migration effect. Now the target model can berestated as:

$\begin{matrix}{{{y(t)} = {{s(t)}*\alpha\;{h_{t}\left( {t - \frac{2\;{R(t)}}{c}} \right)}}},} & (38)\end{matrix}$where the target range changing with time is shown explicitly. In thiscase the range change from pulse-to-pulse must be accounted for in thedetector 590.

As before, eqn. (35) can be approximated by the discrete timeconvolution indicated by:

$\begin{matrix}{{{y_{t}\left( {kT}_{s} \right)} = {\alpha{\sum\limits_{i = 0}^{N - 1}{{s\left( {iT}_{s} \right)}{h_{t}\left( {{\left( {k + 1 - i} \right)T_{s}} - \frac{2\;{R(t)}}{c}} \right)}}}}},} & (39)\end{matrix}$where again the time dependence of R(t) is indicated. This timedependence has the effect of moving the target some significant fraction(or whole) sample times T_(s). While eqn. (36) is accurate, it is notpresented in a convenient form. To address this, the target impulseresponse is redefined as:

$\begin{matrix}{{{h_{t,{R{({\Delta\; T})}}}(t)} = {h_{t}\left( {t - \frac{2\;{R\left( {\Delta\; T} \right)}}{c}} \right)}},} & (40)\end{matrix}$where ΔT is the time since beginning of the CPI. Thus, eqn. (40)accounts for the range of the target and its change over the course ofthe CPI. Next one can show how to determine h_(t.R(ΔT))(t) in terms ofh_(t.R(1))(t). Remembering that h_(t.R(0))(t) is effectively discretetime sampled one can account for the sampling time offset induced by therange change of R(ΔT). This accounts for fractional shifts in the targetposition (as well as whole sample shifts).

Thus, one has:

$\begin{matrix}{{{h_{t,{R{({\Delta\; T})}}}\left( {kT}_{s} \right)} = {\sum\limits_{i}{{h_{t,{R{(0)}}}\left( {lT}_{s} \right)}{{sinc}\left( \frac{\left( {{kT}_{X} - {\Delta\; T_{s}} - {lT}_{s}} \right)}{T_{s}} \right)}}}},} & (41)\end{matrix}$where ΔT_(s)=2[R(ΔT)−R(0)]/c and sin c(x)=sin(πx)/πx. Note that eqn.(41) can be interpreted as the discrete time convolution ofh_(t.R(0))(t) with sin c (t). To determine y_(t)(kT_(s)) for subsequentpulses, one can write:

$\begin{matrix}{{{y_{t}\left( {kT}_{s} \right)} = {\alpha{\sum\limits_{i = 0}^{N - 1}{{s\left( {iT}_{s} \right)}{h_{i,{R{({\Delta\; T})}}}\left( {\left( {k + 1 - i} \right)T_{s}} \right)}}}}},} & (42)\end{matrix}$where ΔT=(m−1)T and in is the pulse number. Next one applies eqn. (41)to give:

$\begin{matrix}{{y_{t}\left( {kT}_{s} \right)} = {\alpha{\sum\limits_{i = 0}^{N - 1}{{s\left( {iT}_{s} \right)}{\sum\limits_{i}{{h_{i,{R{(0)}}}\left( {lT}_{s} \right)}{{{sinc}\left( \frac{\left( {{\left( {k + 1 - i} \right)T_{s}} - {\Delta\; T_{s}} - {l\; T_{s}}} \right)}{T_{s}} \right)}.}}}}}}} & (43)\end{matrix}$where ΔT_(s)=2[R(m−1)T_(i)−R(0)]/c.

Thus, eqn. (43) can be rewritten to change the order of convolution togive time-sampled receive data for one inter-pulse period:

$\begin{matrix}{{y_{t}\left( {kT}_{s} \right)} = {\alpha{\sum\limits_{i}{{h_{i,{R{(0)}}}\left( {lT}_{s} \right)}{\sum\limits_{i = 0}^{N - 1}{{s\left( {iT}_{s} \right)}{{{sinc}\left( \frac{\left( {{\left( {k + 1 - i} \right)T_{s}} - T_{s} - {lT}_{s}} \right)}{T_{s}} \right)}.}}}}}}} & (44)\end{matrix}$With the target model as stated in eqn. (41) enables expressing thetarget model in a manner similar to eqn. (6) as:y _(t) =α{tilde over (R)} _(m)(i){tilde over (S)} _(d) h _(t.R(0),  (45)where h_(t.R(0))=h_(t), {tilde over (S)}_(d) is defined in eqn. (7) and{tilde over (R)}_(m)(m) is the range migration matrix that convolves thesinc( ) function whose elements are determined as:

$\begin{matrix}{{{{\overset{\sim}{R}}_{m}(m)}_{({k,l})} = {{sinc}\left( \frac{\left( {{\left( {k - l} \right)T_{s}} - {\Delta\; T_{s}}} \right)}{T_{s}} \right)}},} & (46)\end{matrix}$where ΔT_(s)=2[R(mT_(i)+lT_(s))−R(0)]/c and m∈{0, . . . , M−1}. Notethat ΔT_(s) is a function of l, the sample number as well as the pulsenumber in. Including lT_(s) accounts for the target motion over the timeof the pulse width.

One can note that h_(t.R(0)) is column vector of size (P+2(N−1))×1,{tilde over (S)}_(d) is rectangular matrix of size (P+N−1)×(P+2(N−1)),and {tilde over (R)}_(m)(m) is a square matrix of size (P+N−1)×(P+N−1).Using eqn. (45) one can write the complete (for all pulses) receivetarget model as:

$\begin{matrix}{Y_{t} = {{\alpha\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}h_{t,{R{(0)}}}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}h_{t,{R{(0)}}}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}h_{t,{R{(0)}}}}\end{bmatrix}}.}} & (47)\end{matrix}$

The detector 590 for fast targets 310, 410 then will be:

$\begin{matrix}{{{{\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}h_{t,{R{(0)}}}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}h_{t,{R{(0)}}}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}h_{t,{R{(0)}}}}\end{bmatrix}^{H}R_{t}^{- 1}Y}}_{\overset{<}{H_{0}}}^{\underset{>}{H_{1}}}\eta},} & (48)\end{matrix}$where the interference correlation matrix R_(l) is defined in eqn. (32)and η is the detection threshold.

Section III—Performance Equations: The processing often done in multiplepulse radars is correlation (i.e., matched filter) in fast time andmoving target detection (MTD) in slow time to integrate returns overslow time. Note that MTD is equivalent to pulse Doppler processing. Thisprocessing is optimal for targets immersed in AWGN. To simplify thisdevelopment, the target acceleration will be ignored (targets areassumed to be constant velocity). Including target acceleration isstraightforward. On a per pulse basis, the output of the correlator atrange cell k given the receive data vector y is:x _(corr)=δ_(k) ^(H) {tilde over (S)} _(d) ^(H) y,  (49)noting that correlator x_(corr) is a complex scalar. The multiplicationby signal convolution matrix Ś_(d) ^(H) performs the operation ofcorrelation on receive data vector y with the transmitted signal vectors. The multiplication by δ_(k) ^(H) sifts out the response of thecorrelator to a target 510 at range cell k. The output of the MTDDoppler filter matched to the target Doppler is:

$\begin{matrix}{{z_{{corr} + {MTD}} = {\alpha\;{u^{H}\begin{bmatrix}x_{{corr}_{1}} \\\vdots \\x_{{corr}_{M}}\end{bmatrix}}}},} & (50)\end{matrix}$where x_(corr) _(t) is the output of the correlator from the i^(th)pulse received.

Subsection (a)—Correlator and Moving Target Detector: The correlatorresponse to a point target for the i^(th) pulse is:x _(corr) _(t) =δ_(k) ^(H) {tilde over (S)} _(d) ^(H) y _(T) _(i) =u_(i)αδ_(k) ^(H) {tilde over (S)} _(d) ^(H) {tilde over (S)}_(d)δ_(k).  (51)Using eqn. (50), the received power due to the target 510 is:

$\begin{matrix}{{z_{{corr} + {MTD}}}^{2} = {{\alpha }^{2}{{{u^{H}\begin{bmatrix}{u_{0}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{\overset{\sim}{S}}_{d}\delta_{k}} \\\vdots \\{u_{M - 1}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{\overset{\sim}{S}}_{d}\delta_{k}}\end{bmatrix}}}^{2}.}}} & (52)\end{matrix}$

On a per pulse basis the response of the correlator to the interferenceis:x _(corr) _(t) =δ_(k) ^(H) {tilde over (S)} _(d) ^(H)({tilde over(S)}c((i−1)T _(i))+n).  (53)

Based on eqn. (50), the output of the Correlator+MTD to the interferenceis:

$\begin{matrix}{{z_{{corr} + {MTD}} = {u^{H}\left( {\left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)\begin{bmatrix}{{\overset{\sim}{S}{c(0)}} + n} \\\vdots \\{{\overset{\sim}{S}{c\left( {\left( {M - 1} \right)T_{i}} \right)}} + n}\end{bmatrix}} \right)}},} & (54)\end{matrix}$where the multiplication by δ_(k) ^(H){tilde over (S)}_(d) ^(H) isunderstood to be done on every element of the of the matrix, where eachelement is in the form of {tilde over (S)}c((M 1)T_(i))|n.

-   -   The power of the interference in the output of Correlator+MTD        is:

$\begin{matrix}{{{E\left\{ {z_{{corr} + {MTD}}}^{2} \right\}} = {E\left\{ {{u^{H}\left( {{{\left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)\begin{bmatrix}{{\overset{\sim}{S}{c(0)}} + n} \\\vdots \\{{\overset{\sim}{S}{c\left( {\left( {M - 1} \right)T_{i}} \right)}} + n}\end{bmatrix}}\begin{bmatrix}{{\overset{\sim}{S}{c(0)}} + n} \\\vdots \\{{\overset{\sim}{S}{c\left( {M - 1} \right)}T_{i}} + n}\end{bmatrix}}^{H}\left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right)} \right)}u} \right\}}},} & (55)\end{matrix}$which can be factored as:

$\begin{matrix}{{E\left\{ {z_{{corr} + {MTD}}}^{2} \right\}} = {{u^{H}\left( {\left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)E\left\{ {\begin{bmatrix}{{\overset{\sim}{S}{c(0)}} + n} \\\vdots \\{{\overset{\sim}{S}{c\left( {\left( {M - 1} \right)T_{i}} \right)}} + n}\end{bmatrix}\begin{bmatrix}{{\overset{\sim}{S}{c(0)}} + n} \\\vdots \\{{\overset{\sim}{S}{c\left( {M - 1} \right)}T_{i}} + n}\end{bmatrix}}^{H} \right\}\left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right)} \right)}{u.}}} & (56)\end{matrix}$By applying eqns. (26) through (32), one has complex voltage output:|z _(corr+MTD)|² =u ^(H)((δ_(k) ^(H) {tilde over (S)} _(d) ^(H))R_(l)({tilde over (S)} _(d)δ_(k)))u.  (57)If the clutter 120 has the same Doppler spectrum for all range cells,then eqn. (33) can be applied giving:|z _(corr+MTD)|² =u ^(H)((δ_(k) ^(H) {tilde over (S)} _(d) ^(H))M _(c)ß{tilde over (S)}R _(c) {tilde over (S)} ^(H)+σ_(n) ² I)({tilde over(S)} _(d)δ_(k)))u.  (58)Moreover, eqn. (58) can be rewritten as:|z _(corr+MTD)|² =u ^(H)((M _(c)ß(δ_(k) ^(H) {tilde over (S)} _(d) ^(H){tilde over (S)}R _(c) {tilde over (S)} ^(H) {tilde over (S)}_(d)δ_(k)))+σ_(n) ² Ißδ _(k) ^(H) {tilde over (S)} _(d) ^(H) {tilde over(S)} _(d)δ_(k))u,  (59)where I is the identity matrix of size M×M.

Applying eqns. (52) and (59), the SIR for the Correlator+MTD can bedetermined as:

$\begin{matrix}{{SIR}_{{corr} + {MTD}} = {\frac{{\alpha }^{2}{{u^{H}\begin{bmatrix}{u_{0}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{\overset{\sim}{S}}_{d}\delta_{k}} \\\vdots \\{u_{M - 1}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{\overset{\sim}{S}}_{d}\delta_{k}}\end{bmatrix}}}^{2}}{{u^{H}\left( {\left( {M_{c} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}\overset{\sim}{S}\; R_{c}{\overset{\sim}{S}}^{H}{\overset{\sim}{S}}_{d}\delta_{k}} \right)} \right) + {\sigma_{n}^{2}{I \otimes \delta_{k}^{H}}{\overset{\sim}{S}}_{d}^{H}{\overset{\sim}{S}}_{d}\delta_{k}}} \right)}u}.}} & (60)\end{matrix}$Next the Correlator+MTD is evaluated for performance against fast movingtargets, i.e., those subject to range migration. First the SIR for theCorrelator+MTD when the detector 590 is not compensating for the rangemigration of the targets. Under this assumption, the target model is:y _(T) _(i) =αu _(i) {tilde over (R)} _(m)(m){tilde over (S)}_(d)δ_(k).  (61)Based on this the output of the Correlator+MTD for a point target is:

$\begin{matrix}{z_{{corr} + {MTD}} = {\alpha\;{{u^{H}\begin{bmatrix}{u_{0}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}(0)}{\overset{\sim}{S}}_{d}\delta_{k}} \\\vdots \\{u_{M - 1}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{\overset{\sim}{S}}_{d}\delta_{k}}\end{bmatrix}}.}}} & (62)\end{matrix}$This in turn gives the power output of the detector 590 due to thetarget 510 as:

$\begin{matrix}{{z_{{corr} + {MTD}}}^{2} = {{\alpha }^{2}{{{u^{H}\begin{bmatrix}{u_{0}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}(0)}{\overset{\sim}{S}}_{d}\delta_{k}} \\\vdots \\{u_{M - 1}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{\overset{\sim}{S}}_{d}\delta_{k}}\end{bmatrix}}}^{2}.}}} & (63)\end{matrix}$Because there is no range migration compensation the interference poweroutput of the Correlator+MTD detector is the same as for the slow targetcase above. Therefore, the SIR for this case is:

$\begin{matrix}{{SIR}_{{corr} + {MTD}} = {\frac{{\alpha }^{2}{{u^{H}\begin{bmatrix}{u_{0}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}(0)}{\overset{\sim}{S}}_{d}\delta_{k}} \\\vdots \\{u_{M - 1}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{\overset{\sim}{S}}_{d}\delta_{k}}\end{bmatrix}}}^{2}}{{u^{H}\left( {\left( {M_{c} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}\overset{\sim}{S}\; R_{c}{\overset{\sim}{S}}^{H}{\overset{\sim}{S}}_{d}\delta_{k}} \right)} \right) + {\sigma_{n}^{2}{I \otimes \delta_{k}^{H}}{\overset{\sim}{S}}_{d}^{H}{\overset{\sim}{S}}_{d}\delta_{k}}} \right)}u}.}} & (64)\end{matrix}$

One can expect that as range migration comes more into play that theCorrelator+MTD is suboptimal therefore degrading the output SIR. This iswhat eqn. (64) enables one to calculate. However, it is straightforwardto modify the Correlator+MTD to account for range migration. Under thiscondition the Correlator+MTD retains it optimality when used in AWGNalone. In order to determine its performance, the SIR for this case willbe derived. The output of the correlator is:corr_(i)=δ_(k) ^(H) {tilde over (S)} _(d) ^(H) αu _(i) {tilde over (R)}′_(m)(m){tilde over (R)} _(m)(m){tilde over (S)} _(d)δ_(k),  (65)where {tilde over (R)}′_(m)(m) is defined the same as {tilde over(R)}_(m)(m) except ΔT_(s)=2(R(0)−R(mT_(i)+kT_(s)))/c. Thusmultiplication by {tilde over (R)}′_(m)(m) compensates for the targetrange migration producing {tilde over (R)}′_(m)(m){tilde over(R)}_(m)(m)≈I. The output of the Correlator+MTD due to the target willbe the same as in the slow target case under the assumption that therange migration has been perfectly compensated. The output of thedetector 590 due to interference changes because the range migrationcompensation affects the clutter 120.

Thus the complex voltage output of the detector 590 due to interferenceis:

$\begin{matrix}{z_{{corr} + {MTD}} = {{u^{H}\left( {\begin{bmatrix}{\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R^{\prime}}}_{m}(0)}\overset{\sim}{S}{c(0)}} \\\vdots \\{\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R^{\prime}}}_{m}\left( {M - 1} \right)}\overset{\sim}{S}{c\left( {\left( {M - 1} \right)T_{i}} \right)}}\end{bmatrix} + n} \right)}.}} & (66)\end{matrix}$This can be rewritten as:

$\begin{matrix}{{z_{{corr} + {MTD}} = {{u^{H}\left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}\left( {\begin{bmatrix}{{{\overset{\sim}{R^{\prime}}}_{m}(0)}\overset{\sim}{S}{c(0)}} \\\vdots \\{{{\overset{\sim}{R^{\prime}}}_{m}\left( {M - 1} \right)}\overset{\sim}{S}{c\left( {\left( {M - 1} \right)T_{i}} \right)}}\end{bmatrix} + n} \right)}},} & (67)\end{matrix}$where one understands that δ_(k) ^(H){tilde over (S)}_(d) ^(H)multiplies each element in the vector where each element is in the formof {tilde over (R)}′_(m)(i−1)c((i−1)T_(i)). From this, the output powerdue to interference will be:

$\begin{matrix}{{E\left\{ {z_{{corr} + {MTD}}}^{2} \right\}} = {u^{H}\left( {\left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)E\left\{ {\left( {\begin{bmatrix}{{{\overset{\sim}{R^{\prime}}}_{m}(0)}\overset{\sim}{S}{c(0)}} \\\vdots \\{{{\overset{\sim}{R^{\prime}}}_{m}\left( {M - 1} \right)}\overset{\sim}{S}{c\left( {\left( {M - 1} \right)T_{i}} \right)}}\end{bmatrix} + n} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {\begin{bmatrix}{{{\overset{\sim}{R^{\prime}}}_{m}(0)}\overset{\sim}{S}{c(0)}} \\\vdots \\{{{\overset{\sim}{R^{\prime}}}_{m}\left( {M - 1} \right)}\overset{\sim}{S}{c\left( {\left( {M - 1} \right)T_{i}} \right)}}\end{bmatrix} + n} \right)^{H}} \right\}\left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right){u.}} \right.}} & (68)\end{matrix}$

To perform the manipulations to get eqn. (71) in a more convenient form,one can simplify notations as follows, c′(i)=c(iT_(i)), where 0≤i≤M−1and defining R_(zc) as:

$\begin{matrix}{R_{zc} = {E{\left\{ {\begin{bmatrix}{{{\overset{\sim}{R^{\prime}}}_{m}(0)}\overset{\sim}{S}{c^{\prime}(0)}} \\\vdots \\{{{\overset{\sim}{R^{\prime}}}_{m}\left( {M - 1} \right)}\overset{\sim}{S}{c^{\prime}\left( {M - 1} \right)}}\end{bmatrix}\begin{bmatrix}{{{\overset{\sim}{R^{\prime}}}_{m}(0)}\overset{\sim}{S}{c^{\prime}(0)}} \\\vdots \\{{{\overset{\sim}{R^{\prime}}}_{m}\left( {M - 1} \right)}\overset{\sim}{S}{c^{\prime}\left( {M - 1} \right)}}\end{bmatrix}}^{H} \right\}.}}} & (69)\end{matrix}$Based on this, the interference power can be written as:E{| _(corr+MTD)|² }=u ^(H)(δ_(k) ^(H) {tilde over (S)} _(d) ^(H))(R_(zc)+σ_(n) I)({tilde over (S)} _(d)δ_(k))u.  (70)Based on this, eqn. (69) can be factored as:

$\begin{matrix}{{R_{zc} = \begin{bmatrix}a_{1,1} & \; & a_{1,2} \\\; & \ddots & \; \\a_{2,1} & \; & a_{2,2}\end{bmatrix}},{where}} & (71) \\\left. \begin{matrix}{{a_{1,1} = {{{\overset{\sim}{R}}_{m}^{\prime}(0)}\overset{\sim}{S}E\left\{ {{c^{\prime}(0)}{c^{\prime\; H}(0)}} \right\}{\overset{\sim}{S}}^{H}{{\overset{\sim}{R}}_{m}^{\prime\; H}(0)}}},} \\{{a_{2,1} = {{{\overset{\sim}{R}}_{m}^{\prime}\left( {M - 1} \right)}\overset{\sim}{S}E\left\{ {{c^{\prime}\left( {M - 1} \right)}{c^{\prime\; H}(0)}} \right\}{\overset{\sim}{S}}^{H}{{\overset{\sim}{R}}_{m}^{\prime\; H}(0)}}},} \\{{a_{1,2} = {{{\overset{\sim}{R}}_{m}^{\prime}(0)}\overset{\sim}{S}E\left\{ {{c^{\prime}(0)}{c^{\prime\; H}\left( {M - 1} \right)}} \right\}{\overset{\sim}{S}}^{H}{{\overset{\sim}{R}}_{m}^{\prime\; H}\left( {M - 1} \right)}}},} \\{{a_{2,2} = {{{\overset{\sim}{R}}_{m}^{\prime}\left( {M - 1} \right)}\overset{\sim}{S}E\left\{ {{c^{\prime}\left( {M - 1} \right)}{c^{\prime\; H}\left( {M - 1} \right)}} \right\}{\overset{\sim}{S}}^{H}{{\overset{\sim}{R}}_{m}^{\prime\; H}\left( {M - 1} \right)}}},}\end{matrix} \right\} & (72)\end{matrix}$and eqn. (28) can be used to determine E{c′(j)c′^(H)(k)}.

Under the assumption that all range cells have the same Doppler spectrumR_(zc) can be simplified to:

$\begin{matrix}{R_{zc} = {\begin{bmatrix}{\rho_{1,1}{{\overset{\sim}{R}}_{m}^{\prime}(0)}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}{R_{m}^{\prime\; H}(0)}} & \; & {\rho_{1,M}{{\overset{\sim}{R}}_{m}^{\prime}(0)}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}{R_{m}^{\prime\; H}\left( {M - 1} \right)}} \\\; & \ddots & \; \\{\rho_{M,1}{{\overset{\sim}{R}}_{m}^{\prime}\left( {M - 1} \right)}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}{R_{m}^{\prime\; H}(0)}} & \; & {\rho_{M,M}{{\overset{\sim}{R}}_{m}^{\prime}\left( {M - 1} \right)}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}{R_{m}^{\prime\; H}\left( {M - 1} \right)}}\end{bmatrix}.}} & (73)\end{matrix}$Using eqns. (57), (74) and (73), the SIR for the Correlator+MTD matchedto the fast target can be written as:

$\begin{matrix}{{{SIR}_{{corr} + {MTD}} = \frac{{\alpha }^{2}A_{1}}{{u^{H}\left( {A_{2} + {\sigma_{n}^{2}{I \otimes \delta_{k}^{H}}{\overset{\sim}{S}}_{d}^{H}{\overset{\sim}{S}}_{d}}} \right)}u}},} & (74)\end{matrix}$where arrays:

$\begin{matrix}{{A_{1} = {{u^{H}\begin{bmatrix}{u_{1}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}^{\prime\;}(0)}{{\overset{\sim}{R}}_{m}(0)}{\overset{\sim}{S}}_{d}\delta_{k}} \\\vdots \\{u_{M - 1}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}^{\prime\;}\left( {M - 1} \right)}{{\overset{\sim}{R}}_{m}^{\;}\left( {M - 1} \right)}{\overset{\sim}{S}}_{d}\delta_{k}}\end{bmatrix}}}^{2}}{and}{A_{2} = \begin{bmatrix}{\rho_{1,1}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}^{\prime\;}(0)}\overset{\sim}{S}{\overset{\sim}{R}}_{c}{\overset{\sim}{S}}^{H}{{\overset{\sim}{R}}_{m}^{\prime\; H}(0)}{\overset{\sim}{S}}_{d}\delta_{k}} & \; & {\rho_{1,M}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}^{\prime\;}(0)}\overset{\sim}{S}{\overset{\sim}{R}}_{c}{\overset{\sim}{S}}^{H}{{\overset{\sim}{R}}_{m}^{\prime\; H}\left( {M - 1} \right)}{\overset{\sim}{S}}_{d}\delta_{k}} \\\; & \ddots & \; \\{\rho_{M,1}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}^{\prime\;}\left( {M - 1} \right)}\overset{\sim}{S}{\overset{\sim}{R}}_{c}{\overset{\sim}{S}}^{H}{{\overset{\sim}{R}}_{m}^{\prime\; H}(0)}{\overset{\sim}{S}}_{d}\delta_{k}} & \; & {\rho_{M,M}\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}{{\overset{\sim}{R}}_{m}^{\prime\;}\left( {M - 1} \right)}\overset{\sim}{S}{\overset{\sim}{R}}_{c}{\overset{\sim}{S}}^{H}{{\overset{\sim}{R}}_{m}^{\prime\; H}\left( {M - 1} \right)}{\overset{\sim}{S}}_{d}\delta_{k}}\end{bmatrix}}} & (75)\end{matrix}$for is pair of array coefficients.

-   -   Subsection (b)—SIR for Optimum Detector for Slow Moving Target:        The detector 590 for the slow moving (no range migration) target        110 is defined in eqn. (34). This detector is optimum for the        assumptions defined including no range migration of the target        and that the interference matrix is known á priori. Here, the        output SIR of the detector 590 is determined. This enables this        detector to be compared to other detectors, as well as the        assessment of performance when the parameters of the signal and        interference processes are not exactly known. In this        development, the target is assumed to be a point target. Under        this assumption, the test statistic from the detector 590 is:        z _(Slow) =u ^(H)ß(δ_(k) ^(H) {tilde over (S)} _(d) ^(H))R _(l)        ^(I) Y.  (76)        From the target model eqn. (24) the received data due to a point        target Y_(T) is:        Y _(T) =αuß{tilde over (S)} _(d)δ_(k).  (77)

The signal power level at the output of the detector 590 is:

$\begin{matrix}{{z_{Slow}}^{2} = {{\alpha }^{2}{{{{u^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{I}^{- 1}{u \otimes {\overset{\sim}{S}}_{d}}\delta_{k}}}^{2}.}}} & (78)\end{matrix}$The output of the detector 590 due to interference is:

$\begin{matrix}{z_{Slow} = {{u^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}{{R_{I}^{- 1}\left( {Y_{c} + Y_{n}} \right)}.}}} & (79)\end{matrix}$The power from the interference is determined as:

$\begin{matrix}{{{E\left\{ {z_{{Slow}_{j}}z_{{Slow}_{j}}^{*}} \right\}} = {E\left\{ {{u^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}{R_{I}^{- 1}\left( {Y_{c} + Y_{n}} \right)}\left( {Y_{c} + Y_{n}} \right)^{H}R_{I}^{- 1}{u \otimes \left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right)}} \right\}}},} & (80)\end{matrix}$which can be factored into:

$\begin{matrix}{{E\left\{ {z_{{Slow}_{j}}z_{{Slow}_{j}}^{*}} \right\}} = {{u^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{I}^{- 1}E\left\{ {\left( {Y_{c} + Y_{n}} \right)\left( {Y_{c} + Y_{n}} \right)^{*}} \right\} R_{I}^{- 1}{u \otimes {\left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right).}}}} & (81)\end{matrix}$Recognizing that the interference correlation matrix:

$\begin{matrix}{{R_{I} = {E\left\{ {\left( {Y_{c} + Y_{n}} \right)\left( {Y_{c} + Y_{n}} \right)^{*}} \right\}}},} & (82)\end{matrix}$enables simplifying eqn. (62) to:

$\begin{matrix}{{E\left\{ {z_{{Slow}_{j}}z_{{Slow}_{j}}^{*}} \right\}} = {{u^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{j}^{- 1}{u \otimes {\left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right).}}}} & (83)\end{matrix}$This gives the SIR for the slow detector as:

$\begin{matrix}{{SIR}_{{corr} + {MTD}} = {\frac{{\alpha }^{2}{{{u^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{I}^{- 1}{u \otimes \left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right)}}}^{2}}{{u^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{I}^{- 1}{u \otimes \left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right)}}.}} & (84)\end{matrix}$which can be simplified to:

$\begin{matrix}{{SIR}_{{corr} + {MTD}} = {{\alpha }^{2}{{{u^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{I}^{- 1}{u \otimes {\left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right).}}}}}} & (85)\end{matrix}$

Next the SIR for a mismatched detector will be determined. To begin thetarget pulse-to-pulse phase change vector used by the detector 590 isdesignated u_(D) while the detector's interference correlation matrix issimilar to eqn. (33):

$\begin{matrix}{{R_{I_{a}} = {{{M_{c_{D}} \otimes \overset{\sim}{S}}R_{c_{D}}{\overset{\sim}{S}}^{H}} + {\sigma_{n_{D}}^{2}I}}},} & (86)\end{matrix}$where the subscript D denotes the detector's assumed parameter. In asimilar manner target's actual pulse-to-pulse phase change vector isdesignated u_(D) and the actual interference correlation matrix is:

$\begin{matrix}{R_{I_{d}} = {{{M_{c_{d}} \otimes \overset{\sim}{S}}R_{c_{d}}{\overset{\sim}{S}}^{H}} + {\sigma_{n_{d}}^{2}{I.}}}} & (87)\end{matrix}$The subscript A denoting the actual value of the parameter. Based oneqn. (76), the detector test statistic becomes:

$\begin{matrix}{z_{Slow} = {{{u_{D}^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}\left( {{{M_{c_{D}} \otimes \overset{\sim}{S}}R_{c_{D}}{\overset{\sim}{S}}^{H}} + {\sigma_{n_{D}}^{2}I}} \right)^{- 1}Y} = {{u_{D}^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{l_{D}}^{- 1}{Y.}}}} & (88)\end{matrix}$The input to the detector 590 from the target 510 is:

$\begin{matrix}{Y_{T} = {\alpha\;{u_{A} \otimes {\left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right).}}}} & (89)\end{matrix}$

Applying this to eqn. (76) gives the mismatched detector's response tothe target 510 as:

$\begin{matrix}{z_{{Slow}_{r}} = {{u_{D}^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{I_{p}}^{- 1}\alpha\;{u_{A} \otimes {\left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right).}}}} & (90)\end{matrix}$The power due to the target in the mismatched detector is:

$\begin{matrix}{{z_{{Slow}_{r}}}^{2} = {{\alpha }^{2}{{{{u_{D}^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{I_{b}}^{- 1}{u_{A} \otimes \left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right)}}}^{2}.}}} & (91)\end{matrix}$In a similar manner, the interference induced in the mismatched slowdetector can be shown to be:

$\begin{matrix}{\left. {E\left\{ z_{{Slow}_{r}} \right.^{2}} \right\} = {{u_{D}^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{I_{D}}^{- 1}R_{I_{A}}R_{I_{D}}^{- 1}{u_{D} \otimes {\left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right).}}}} & (92)\end{matrix}$This give the SIR for the mismatched detector as:

$\begin{matrix}{{SIR}_{{corr} + {MTD}} = {\frac{{\alpha }^{2}{{{u_{D}^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{I_{o}}^{- 1}{u_{A} \otimes \left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right)}}}^{2}}{{u_{D}^{H} \otimes \left( {\delta_{k}^{H}{\overset{\sim}{S}}_{d}^{H}} \right)}R_{I_{D}}^{- 1}{u_{D} \otimes \left( {{\overset{\sim}{S}}_{d}\delta_{k}} \right)}}.}} & (93)\end{matrix}$

Subsection (c)—SIR for Optimum Detector for Fast Moving Target: Thedetector 590 is presented in eqn. (48). Here, the target will be assumedto be a point target. The test statistic is computed as:

$\begin{matrix}{z_{Fast} = {\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H}R_{I}^{- 1}{Y.}}} & (94)\end{matrix}$From the target model eqn. (44) the power output of the detector 590 dueto the target 510 is:

$\begin{matrix}{{z_{{Fast}_{I}}}^{2} = {{\alpha }^{2}{{{\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H}{R_{I}^{- 1}\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}}\; }^{2}.}}} & (95)\end{matrix}$The output of the detector 590 due to the interference is:

$\begin{matrix}{z_{{Fast}_{I}} = {\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H}R_{I}^{- 1}{Y_{I}.}}} & (96)\end{matrix}$The power of the interference is found by taking the expectation of|z_(Fast) ₁ |² giving:

$\begin{matrix}{{z_{{Fast}_{I}}}^{2} = {E{\left\{ {\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H}R_{I}^{- 1}Y_{I}Y_{I}{R_{I}^{- 1}\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}}\; \right\}.}}} & (97)\end{matrix}$This can be factored to produce:

$\begin{matrix}{{z_{{Fast}_{I}}}^{2} = {\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H}R_{I}^{- 1}E\left\{ {Y_{I}Y_{I}^{H}} \right\}{{R_{I}^{- 1}\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}\;.}}} & (98)\end{matrix}$which can be simplified to:

$\begin{matrix}{{z_{{Fast}_{I}}}^{2} = {\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H}{{R_{I}^{- 1}\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}\;.}}} & (99)\end{matrix}$

Combining eqns. (95) and (99) enables computing the SIR for the Fastdetector as:

$\begin{matrix}{{{SIR}_{Fast}{\alpha }^{2}} = {\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H}{{R_{I}^{- 1}\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}\;.}}} & (100)\end{matrix}$Next the SIR for a mismatched detector will be determined. To begin,u_(D) designates the target pulse-to-pulse phase change vector used bythe detector 590, {tilde over (R)}_(m) _(D) is the target rangemigration matrix used by the detector 590, and R_(l) _(D) is thedetector's interference correlation matrix as defined in eqn. (86). Thesubscript D denotes the detector's assumed parameter. In a similarmanner, u_(A) designates the target's actual pulse-to-pulse phase changevector, {tilde over (R)}_(m) _(A) is the actual target range migrationmatrix, and R_(l) _(A) is the actual interference correlation matrix asdefined in eqn. (87). The subscript A indicates the actual value.

The power output of the mismatched filter due to the target 510 is:

$\begin{matrix}{{z_{{Fast}_{I}}}^{2} = {{{\alpha }^{2}\begin{bmatrix}{u_{D_{0}}{{\overset{\sim}{R}}_{m_{D}}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{D_{1}}{{\overset{\sim}{R}}_{m_{D}}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{D_{M - 1}}{{\overset{\sim}{R}}_{m_{D}}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}^{H}{{R_{I_{D}}^{- 1}\begin{bmatrix}{u_{A_{0}}{{\overset{\sim}{R}}_{m_{A}}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{A_{1}}{{\overset{\sim}{R}}_{m_{A}}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{A_{M - 1}}{{\overset{\sim}{R}}_{m_{A}}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}\;.}}} & (101)\end{matrix}$To determine the mismatched filter interference power eqn. (98) can berewritten as:

$\begin{matrix}{{z_{{Fast}_{I}}}^{2} = {\begin{bmatrix}{u_{D_{0}}{{\overset{\sim}{R}}_{m_{D}}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{D_{1}}{{\overset{\sim}{R}}_{m_{D}}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{D_{M - 1}}{{\overset{\sim}{R}}_{m_{D}}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H}R_{I_{D}}^{- 1}E\left\{ {Y_{I}Y_{I}^{H}} \right\}{{R_{I_{D}}^{- 1}\begin{bmatrix}{u_{D_{0}}{{\overset{\sim}{R}}_{m_{D}}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{D_{1}}{{\overset{\sim}{R}}_{m_{D}}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{D_{M - 1}}{{\overset{\sim}{R}}_{m_{D}}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}\;.}}} & (102)\end{matrix}$which can be simplified to:

$\begin{matrix}{{z_{{Fast}_{I}}}^{2} = {\begin{bmatrix}{u_{D_{0}}{{\overset{\sim}{R}}_{m_{D}}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{D_{1}}{{\overset{\sim}{R}}_{m_{D}}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{D_{M - 1}}{{\overset{\sim}{R}}_{m_{D}}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H}R_{I_{D}}^{- 1}R_{I_{A}}{{R_{I_{D}}^{- 1}\begin{bmatrix}{u_{A_{0}}{{\overset{\sim}{R}}_{m_{d}}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{A_{1}}{{\overset{\sim}{R}}_{m_{d}}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{A_{M - 1}}{{\overset{\sim}{R}}_{m_{d}}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}.}}} & (103)\end{matrix}$The SIR for the Fast mismatched filter is obtained by dividing eqn.(101) by eqn. (103) to produce:

$\begin{matrix}{{{SIR}_{{Fast}_{mismatched}} = \frac{\begin{matrix}{{\alpha }^{2}{\begin{bmatrix}{u_{D_{0}}{{\overset{\sim}{R}}_{m_{D}}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{D_{1}}{{\overset{\sim}{R}}_{m_{D}}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{D_{M - 1}}{{\overset{\sim}{R}}_{m_{D}}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H}}} \\{{R_{I_{D}}^{- 1}\begin{bmatrix}{u_{A_{0}}{{\overset{\sim}{R}}_{m_{s}}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{A_{1}}{{\overset{\sim}{R}}_{m_{s}}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{A_{M - 1}}{{\overset{\sim}{R}}_{m_{s}}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}}^{2}\end{matrix}}{\begin{matrix}\begin{bmatrix}{u_{D_{0}}{{\overset{\sim}{R}}_{m_{D}}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{D_{1}}{{\overset{\sim}{R}}_{m_{D}}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{D_{M - 1}}{{\overset{\sim}{R}}_{m_{D}}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}^{H} \\{R_{I_{D}}^{- 1}R_{I_{A}}{R_{I_{D}}^{- 1}\begin{bmatrix}{u_{D_{0}}{{\overset{\sim}{R}}_{m_{D}}(0)}{{\overset{\sim}{S}}_{d}(0)}\delta_{k}} \\{u_{D_{1}}{{\overset{\sim}{R}}_{m_{D}}(1)}{{\overset{\sim}{S}}_{d}(1)}\delta_{k}} \\\vdots \\{u_{D_{M - 1}}{{\overset{\sim}{R}}_{m_{D}}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}\delta_{k}}\end{bmatrix}}}\end{matrix}}},} & (104)\end{matrix}$

Subsection (d)—Detectors for Multiple Non-Identical Pulses: Here, theoptimum detector 590 for the case that multiple pulses are transmittedcoherently and it is desired to maximize detection probability for agiven probability of false alarm in the presence of clutter 120.Normally radars 130 transmit identical pulses. However, recentliterature has shown how that the pulses may be different subject toconstraints. This approach removes these constraints while producing anoptimum detector (subject to the previously imposed assumptions).

In this development, one can assume that range migration can be ignored,i.e., the eqn. (1) rate inequality, that the target acceleration can beignored and the target is a point target, for simplicity of notation.Note that including range migration, target acceleration and finiteextent into this formulation is straightforward using the previousdeveloped approach. Each pulse transmitted is s_(i) which is a vectorcontaining the baseband samples of the i^(th) pulse that is transmitted.This produces the respective signal convolution matrices as:

$\begin{matrix}{{\overset{\sim}{S}}_{i} = {\begin{bmatrix}0 & \ldots & 0 & s_{i}^{t} \\0 & \ldots & s_{i}^{t} & 0 \\\; & \ddots & \; & \; \\s_{i}^{i} & 0 & \ldots & 0\end{bmatrix}.}} & (105)\end{matrix}$

The received data due to a point target are then matrix:

$\begin{matrix}{Y_{T} = {{\alpha\begin{bmatrix}{u_{0}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{1}{\overset{\sim}{S}}_{2,d}\delta_{k}} \\\vdots \\{u_{M - 1}{\overset{\sim}{S}}_{M,d}\delta_{k}}\end{bmatrix}}.}} & (106)\end{matrix}$In a similar manner, the received data due to clutter 120 will besimilar to eqn. (25):

$\begin{matrix}{Y_{c} = {\begin{bmatrix}{{\overset{\sim}{S}}_{1}{c(0)}} \\{{\overset{\sim}{S}}_{2}{c\left( {\left( {i - 1} \right)T_{i}} \right)}} \\\vdots \\{{\overset{\sim}{S}}_{M}{c\left( {\left( {M - 1} \right)T_{i}} \right)}}\end{bmatrix}.}} & (107)\end{matrix}$

From this the clutter correlation can be determined as:

$\begin{matrix}{R_{y_{c}} = {E{\left\{ {\begin{bmatrix}{{\overset{\sim}{S}}_{1}{c(0)}} \\{{\overset{\sim}{S}}_{2}{c\left( T_{i} \right)}} \\\vdots \\{{\overset{\sim}{S}}_{M}{c\left( {\left( {M - 1} \right)T_{i}} \right)}}\end{bmatrix}\left\lbrack {{c^{H}(0)}{\overset{\sim}{S}}_{1}^{H}\mspace{14mu}{c^{H}\left( T_{1} \right)}{{\overset{\sim}{S}}_{2}^{H}\;.\;.\;.\;{c^{H}\left( {\left( {M - 1} \right)T_{1}} \right)}}{\overset{\sim}{S}}_{M}^{H}} \right\rbrack} \right\}.}}} & (108)\end{matrix}$Under the assumption that the clutter 120 has the same Doppler spectrumthe clutter correlation matrix can be written similar to eqn. (30) as:

$\begin{matrix}{R_{y_{c}} = {\begin{bmatrix}{\rho_{1,1}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}} & \; & {\rho_{1,M}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}} \\\; & \ddots & \; \\{\rho_{M,1}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}} & \; & {\rho_{M,M}\overset{\sim}{S}R_{c}{\overset{\sim}{S}}^{H}}\end{bmatrix}.}} & (109)\end{matrix}$Clearly the noise correlation matrix is unchanged. Therefore, it isstraightforward to write the optimum detector 590 for non-identicalpulses as:

$\begin{matrix}{{{{\begin{bmatrix}{u_{0}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{1}{\overset{\sim}{S}}_{2,d}\delta_{k}} \\\vdots \\{u_{M - 1}{\overset{\sim}{S}}_{M,d}\delta_{k}}\end{bmatrix}^{H}R_{I}^{- 1}Y}}\begin{matrix}\overset{H_{1}}{>} \\\underset{H_{0}}{<}\end{matrix}\eta},} & (110)\end{matrix}$where the interference correlation matrix in similarity to eqn. (37) is:

$\begin{matrix}{R_{I} = {\left\lbrack {\begin{bmatrix}{\rho_{1,1}{\overset{\sim}{S}}_{1}R_{c}{\overset{\sim}{S}}_{1}^{H}} & \; & {\overset{\sim}{S}\rho_{1,M}{\overset{\sim}{S}}_{1}R_{c}{\overset{\sim}{S}}_{M}^{H}} \\\; & \ddots & \; \\{\rho_{M,1}{\overset{\sim}{S}}_{M}R_{c}{\overset{\sim}{S}}_{1}^{H}} & \; & {\rho_{M,M}{\overset{\sim}{S}}_{M}R_{c}{\overset{\sim}{S}}_{M}^{H}}\end{bmatrix} + {\sigma_{n}^{2}I}} \right\rbrack.}} & (111)\end{matrix}$

Subsection (e)—SIR for Non-Identical Pulse Detector: The power output ofthe detector 590 due to the signal is:

$\begin{matrix}{{z_{r}}^{2} = {{{\alpha }^{2}\left\lbrack {\begin{bmatrix}{u_{0}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{0}{\overset{\sim}{S}}_{2,d}\delta_{k}} \\\vdots \\{u_{M - 1}{\overset{\sim}{S}}_{M,d}\delta_{k}}\end{bmatrix}^{H}{R_{I}^{- 1}\begin{bmatrix}{u_{0}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{0}{\overset{\sim}{S}}_{2,d}\delta_{k}} \\\vdots \\{u_{0}{\overset{\sim}{S}}_{M,d}\delta_{k}}\end{bmatrix}}} \right\rbrack}^{2}.}} & (112)\end{matrix}$The power output of the of the detector 590 due to the interference is:

$\begin{matrix}{{z_{I}}^{2} = {\begin{bmatrix}{u_{0}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{1}{\overset{\sim}{S}}_{2,d}\delta_{k}} \\\vdots \\{u_{M - 1}{\overset{\sim}{S}}_{M,d}\delta_{k}}\end{bmatrix}^{H}{{R_{I}^{- 1}\begin{bmatrix}{u_{0}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{1}{\overset{\sim}{S}}_{2,d}\delta_{k}} \\\vdots \\{u_{M - 1}{\overset{\sim}{S}}_{M,d}\delta_{k}}\end{bmatrix}}.}}} & (113)\end{matrix}$This produces the SIR for the Non-Identical pulse detector as:

$\begin{matrix}{{SIR}_{NIP} = {\begin{bmatrix}{u_{0}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{1}{\overset{\sim}{S}}_{2,d}\delta_{k}} \\\vdots \\{u_{M - 1}{\overset{\sim}{S}}_{M,d}\delta_{k}}\end{bmatrix}^{H}{{R_{I}^{- 1}\begin{bmatrix}{u_{0}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{1}{\overset{\sim}{S}}_{2,d}\delta_{k}} \\\vdots \\{u_{M - 1}{\overset{\sim}{S}}_{M,d}\delta_{k}}\end{bmatrix}}.}}} & (114)\end{matrix}$

Next the mismatched detector 590 SIR is determined. The designpulse-to-pulse phase change vector is u_(D), the actual pulse-to-pulsephase change vector is u_(A). The design interference correlation matrixis:

$\begin{matrix}{R_{I_{D}} = {\begin{bmatrix}{\rho_{D,1,1}{\overset{\sim}{S}}_{1}R_{c_{D}}{\overset{\sim}{S}}_{1}^{H}} & \; & {\rho_{D,1,M}{\overset{\sim}{S}}_{1}R_{c_{D}}{\overset{\sim}{S}}_{M}^{H}} \\\; & \ddots & \; \\{\rho_{D,M,1}{\overset{\sim}{S}}_{M}R_{c_{D}}{\overset{\sim}{S}}_{1}^{H}} & \; & {\rho_{D,M,M}{\overset{\sim}{S}}_{M}R_{c_{D}}{\overset{\sim}{S}}_{M}^{H}}\end{bmatrix} + {\sigma_{n_{D}}{I.}}}} & (115)\end{matrix}$Similarly, the actual interference correlation matrix is:

$\begin{matrix}{R_{I_{A}} = {\begin{bmatrix}{\rho_{A,1,1}{\overset{\sim}{S}}_{1}R_{c_{A}}{\overset{\sim}{S}}_{1}^{H}} & \; & {\rho_{A,1,M}{\overset{\sim}{S}}_{1}R_{c_{A}}{\overset{\sim}{S}}_{M}^{H}} \\\; & \ddots & \; \\{\rho_{A,M,1}{\overset{\sim}{S}}_{M}R_{c_{A}}{\overset{\sim}{S}}_{1}^{H}} & \; & {\rho_{A,M,M}{\overset{\sim}{S}}_{M}R_{c_{A}}{\overset{\sim}{S}}_{M}^{H}}\end{bmatrix} + {\sigma_{n_{i}}{I.}}}} & (116)\end{matrix}$From this, the SIR for the mismatched NIP detector 590 can be determinedas:

$\begin{matrix}{{SIR}_{{NIP}_{mismatched}} = {\frac{{{\alpha }^{2}\left\lbrack {\begin{bmatrix}{u_{D_{0}}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{D_{1}}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\\vdots \\{u_{D_{M - 1}}{\overset{\sim}{S}}_{1,d}\delta_{k}}\end{bmatrix}^{H}{R_{I_{D}}\begin{bmatrix}{u_{A_{0}}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{A_{1}}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\\vdots \\{u_{A_{M - 1}}{\overset{\sim}{S}}_{1,d}\delta_{k}}\end{bmatrix}}} \right\rbrack}^{2}}{\begin{bmatrix}{u_{D_{0}}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{D_{1}}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\\vdots \\{u_{D_{M - 1}}{\overset{\sim}{S}}_{1,d}\delta_{k}}\end{bmatrix}^{H}R_{I_{D}}^{- 1}R_{I_{s}}{R_{I_{A}}^{- 1}\begin{bmatrix}{u_{D_{0}}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\{u_{D_{1}}{\overset{\sim}{S}}_{1,d}\delta_{k}} \\\vdots \\{u_{D_{M - 1}}{\overset{\sim}{S}}_{1,d}\delta_{k}}\end{bmatrix}}}.}} & (117)\end{matrix}$

Section IV—Clutter Motion: Here, the effect of radar motion isincorporated into the detector 590. In the preceding developments themotion of the clutter due was ignored. For clutter motion that does notcause range migration, the induced clutter motion can be handled anumber of documented techniques. This analysis deals with rangemigrating clutter as well as clutter that may have different meanDoppler in reach range cell.

Subsection (a)—Range Mitigating Clutter: Radars 130 mounted on platformstraveling at high speeds (such as aircraft or satellites) produces rangemigration effects on the received clutter signal as well as a Doppleroffset due the radar motion. To begin with the mean Doppler for groundreflections is:

$\begin{matrix}{{f_{d} = {{\frac{2v_{r}}{\lambda}{\cos\left( \theta_{cone} \right)}} = {\frac{2v_{r}}{\lambda}{\cos\left( \theta_{az} \right)}{\cos\left( \theta_{el} \right)}}}},} & (118)\end{matrix}$where v_(r) is the radar velocity, θ_(cone) is the cone angle, θ_(az) isthe azimuth angle and θ_(el) is the elevation angle. The angles arerelative to the radar velocity vector and a specific location on theground.

To illustrate how range migration compensation can be applied the simplecase of the radar traveling north at a constant velocity of v_(r) at aconstant altitude over a flat earth will be used. The techniquedeveloped however, is not limited to the simple geometry. Instead, thismethod is used to simplify the illustrations. The coordinate system usedis a follows. The radar height above the ground is γ_(Z) _(R) , theradar east/west position is γ_(X) _(R) is a constant zero, and the radarnorth/south position is γ_(Y) _(R) =ν_(r)t indicating that at time zero(the beginning of the CPI) the radar's position is γ_(X) _(R) =0, γ_(Y)_(R) =0 and γ_(Z) _(R) =constant.

FIGS. 6A, 6B and 6C show plan graphical views 600 of plots 605, 610 and615 looking downward with directions east 620 as the abscissa and north625 as the ordinate. The radar 130 travels in the direction of north625, searching from start position 630 that corresponds to the beginningof CPI at an antenna start pointing angle 635. Plot 605 depicts a beamvector 640 with minimum range arc 645, a Ground Reference Point (GRP)650, resolution cell 655, clutter region 660 and maximum range arc 665.

From the start position 630, the radar 130 emits a transmit signal alongthe beam vector 640 across a beam width range denoted by arcs 645 and665. Plot 610 shows a series of GRPs 650 along the beam vector 640within an arc region 670 of discernable clutter. Plot 615 shows effectof radar motion from start position 630 at the antenna start pointingangle to end position 680 of CPI at the antenna end pointing angle 685.From the end position 680, the radar 130 emits a transmit signal along abeam vector 690 and along beam width 695. The GRP 650 corresponds to theintersection of the vectors 640 and 690.

The spatial relationship of the radar relative to the stationary groundclutter 120 is illustrated in plan views 600, which hides the altitudeand elevation aspect of the problem for simplicity. Graph 605 indicatesthe minimum and maximum range of the radar 130 as respective arcs 645and 665. The ranges are determined by the turning on and off the radarreceiver 520 on a per-pulse basis. The beam width 695 of the radarantenna 535 (as projected to the ground) is shown as well. These limitsdefine the region of the ground that produces discernable clutter 120that interferes with target detection. Also indicated in graph 605 isthe radar resolution cell 655 projected on the earth's surface. This isthe spatial region 660 that produces clutter 120 for receive sample y.At the center of this region is the GRP 650, which provides a referencepoint to determine mean Doppler frequency eqn. (118) and range migrationof the clutter 120.

Consequently, each range resolution cell has an associated GRP 650 asillustrated in graph 610. From each GRP 650 in each cell 655, the meanDoppler and corresponding range change can be calculated. There areP+2(N−1) GRPs 650 corresponding to the P+2(N−1) elements of the vector cpreviously defined. As the radar 130 moves over the time period of theCPI, the range to the GRP 650 changes as well as its angle, asillustrated in view 615. The range to a given GRP 650 is:

$\begin{matrix}{{R_{GRP}(t)} = {\sqrt{\gamma_{X_{GRP}}^{2} + \left( {\gamma_{X_{GRP}} - {v_{r}t}} \right)^{2} + \gamma_{Z_{GRP}}^{2}}.}} & (119)\end{matrix}$The azimuth angle is:

$\begin{matrix}{{\theta_{az} = {\sin^{- 1}\left( \frac{\gamma_{X_{GRP}}}{\sqrt{\left( {\gamma_{Y_{GRP}} - {v_{r}t}} \right)^{2} + \gamma_{X_{GRP}}^{2}}} \right)}},} & (120)\end{matrix}$and the elevation angle is:

$\begin{matrix}{{\theta_{el} = {\sin^{- 1}\left( \frac{\gamma_{Z_{GRO}}}{\sqrt{\left( {\gamma_{Y_{GRP}} - {v_{r}t}} \right)^{2} + \gamma_{X_{GRP}}^{2}}} \right)}},} & (121)\end{matrix}$where γ_(X) _(GRP) γ_(Y) _(GRP) and γ_(Z) _(GRP) are the coordinates ofthe GRP 650.

The cone angle of the GRP 650 is:

$\begin{matrix}{{\theta_{cone}(t)} = {{\cos^{- 1}\left( {{\cos\left( {\theta_{az}(t)} \right)}{\cos\left( {\theta_{el}(t)} \right)}} \right)}.}} & (122)\end{matrix}$The range rate is:

$\begin{matrix}{{{\overset{.}{R}}_{RGP}^{i}(t)} = {\frac{{dR}_{GRP}(t)}{dt} = {\frac{{{- \gamma_{Y_{GRP}}}v_{r}} + {v_{r}^{2}t}}{\sqrt{\gamma_{X_{GRP}}^{2} + \left( {\gamma_{Y_{GRP}} - {v_{r}t}} \right)^{2} + \gamma_{Z_{g}}^{2}}}.}}} & (123)\end{matrix}$Next it is convenient to rewrite range to GRP R_(GRP)(t) as a functionof the initial range of the GRP 650 and the initial angles 635. Startingwith:

$\begin{matrix}{{{R_{GRP}^{2}(t)} = {{\gamma_{X_{GRP}}^{2} + \left( {\gamma_{X_{GRP}} - {v_{r}t}} \right)^{2} + \gamma_{Z_{GRP}}^{2}} = {{R_{GRP}^{2}(0)} + {v_{r}^{2}t^{2}} - {2\gamma_{Y_{GRP}}v_{r}t}}}},} & (124)\end{matrix}$and then substituting north/south position of the GRP 650:

$\begin{matrix}{{\gamma_{Y_{GRP}} = {{R_{GRP}(0)}{\cos\left( \theta_{el} \right)}{\sin\left( \theta_{az} \right)}}},} & (125)\end{matrix}$produces the range:

$\begin{matrix}{{R_{GRP}(t)} = {\sqrt{{R_{GRP}^{2}(0)} + {v_{r}^{2}t^{2}} - {2{R_{GRP}(0)}{\cos\left( \theta_{el} \right)}{\sin\left( \theta_{az} \right)}v_{r}t}}.}} & (126)\end{matrix}$

As stated earlier, the range extent of the clutter 120 is determined bythe radar receiver switching time relative to the beginning of thetransmit pulse time. If the receiver turn on delay time is (for eachreceive interval) is T_(rx delay) then the range of the closest GRP 650is cT_(rx_delay)/2. From this the range of each GRP 650 is determinedas:

$\begin{matrix}{{{R_{GRP}^{\prime}(0)} = \frac{\left( {T_{rx\_ delay} + {\left( {i - 1} \right)T_{s}}} \right)c}{2}},} & (127)\end{matrix}$where there are P+2(N−1) GRPs 650 and i∈1, . . . , P+(N−1). Because theradar 130 is in motion and the GRPs 650 are stationary, the GRP rangeschange over the course of the CPI. The range as a function of time foreach GRP 650 can be written as:

$\begin{matrix}{{R_{GRP}^{\prime}(t)} = {\sqrt{\begin{matrix}{{\frac{c}{2}\left( {T_{rx\_ delay} + {\left( {i - 1} \right)T_{x}}} \right)^{2}} + {v_{r}^{2}t^{2}} - {2\left( \frac{c}{2} \right)\left( {T_{rx\_ delay} +} \right.}} \\{\left. {\left( {i - 1} \right)T} \right){\cos\left( \theta_{el} \right)}{\sin\left( \theta_{az} \right)}}\end{matrix}}.}} & (128)\end{matrix}$Note that the range rate {dot over (R)}_(RGP) ^(i)(t) will also varywith i. The amount of change depends on the geometry of the situation.In many cases, the range migration for clutter 120 will be substantiallythe same for each GRP 650. This is stated mathematically as:

$\begin{matrix}{{{{{R_{RGP}^{i}\left( {MT}_{i} \right)} - {R_{GRP}^{i}(0)}}} - {{{{R_{RGP}^{i}\left( {MT}_{i} \right)} - {R_{GRP}^{i}(0)}}}{\operatorname{<<}\frac{c}{2}}T_{x}}},{\forall i},{j.}} & (129)\end{matrix}$In Subsection (b), the detector 590 is developed under the assumptionthat clutter range migration is the same for all range resolution cells655.

Subsection (b)—Uniform Range Migration Detector: The detector 590 thataccommodates target range migration and clutter range migration will bedeveloped. This detector 590 is optimum subject to the previously statedassumptions. Further, one can assume that the range migration is thesame for each range cell 655, i.e., eqn. (129) is satisfied. Subsection(c) will handle the general case of clutter range migration beingdifferent for each range cell 655. Because the range rate for each GRP650 can be different, Doppler frequency f_(d) will be different for theclutter 120 that arises from each resolution cell 655.

This is accommodated by reformulating the signal convolution matrix in asimilar manner to {tilde over (S)}_(d)(m) that was formulated to accountfor the target Doppler. Thus, the signal convolution matrix for theclutter {tilde over (S)}_(c)(m) is defined as:

$\begin{matrix}{{{{\overset{\sim}{S}}_{c}(m)} = \begin{bmatrix}0 & \; & \ldots & {s_{1}e^{j\;{\Delta_{N}{({m,1})}}}} & \ldots & {s_{N}e^{j\;{\Delta_{i}{({m,N})}}}} \\\; & \; & ⋰ & \; & \; & \; \\{s_{1}e^{j\;{\Delta\;}_{p + {2{({N - 1})}}}{({m,1})}}} & \ldots & {s_{N}e^{j\;{\Delta_{\mu,{2{{({N - 1})} \cdot {({N - 1})}}}}{({m,N})}}}} & \ldots & \; & 0\end{bmatrix}},} & (130)\end{matrix}$where Δ_(i)(m,n) is the phase change of the clutter 120 at the i^(th)GRP 650 from the beginning of the CPI at position 635 to the currentpulse n and sample n and is calculated as:

$\begin{matrix}{{\Delta_{i}\left( {m,n} \right)} = {\frac{4\pi}{\lambda}{\left( {{R_{GRP}^{i}\left( {{mT}_{i} + {\left( {n - 1} \right)T_{x}}} \right)} - {R_{GRP}^{i}(0)}} \right).}}} & (131)\end{matrix}$

Under the assumption that range migration is the same for each rangecell 655, there is a single range migration matrix. Therefore, the rangemigration matrix defined in eqn. (46) with ΔT_(s) calculated as:

$\begin{matrix}{{\Delta\; T_{s}} = {\frac{2}{c}{\left( {{R_{GRP}\left( {{mT}_{i} + {IT}_{g}} \right)} - {R_{RGP}(0)}} \right).}}} & (132)\end{matrix}$The receive data due to clutter 120 are:

$\begin{matrix}{Y_{c} = {\begin{bmatrix}{{{\overset{\sim}{R}}_{m}(0)}{{\hat{S}}_{c}(0)}{c(0)}} \\\vdots \\{{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\hat{S}}_{c}\left( {M - 1} \right)}{c\left( {M - 1} \right)}}\end{bmatrix}.}} & (133)\end{matrix}$

The clutter correlation matrix is determined as:

$\begin{matrix}{{R_{Y_{c}} = {{E\left\{ {Y_{c}Y_{c}^{H}} \right\}} = {E\left\{ {\begin{bmatrix}{{{\overset{\sim}{R}}_{m}(0)}{{\hat{S}}_{c}(0)}{c(0)}} \\\vdots \\{{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\hat{S}}_{c}\left( {M -} \right.}} \\{\left. 1 \right){c\left( {M - 1} \right)}}\end{bmatrix}\begin{bmatrix}{{{\overset{\sim}{R}}_{m}(0)}{{\hat{S}}_{c}(0)}{c(0)}} \\\vdots \\{{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\hat{S}}_{c}\left( {M -} \right.}} \\{\left. 1 \right){c\left( {M - 1} \right)}}\end{bmatrix}}^{H} \right\}}}},} & \square\end{matrix}$which can be written as:

$\begin{matrix}{{R_{Y_{c}} = \begin{bmatrix}a_{1,1} & \; & a_{1,2} \\\; & \ddots & \; \\a_{2,1} & \; & a_{2,2}\end{bmatrix}},{where}} & (135) \\\left. \begin{matrix}{{a_{1,1} = {{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{c}(0)}\rho_{1,1}R_{c}{{\overset{\sim}{S}}_{c}^{H}(0)}{{\overset{\sim}{R}}_{m}^{\prime\; H}(0)}}},} \\{{a_{1,2} = {{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{c}(0)}\rho_{M,1}R_{c}{{\overset{\sim}{S}}_{c}^{H}\left( {M - 1} \right)}{{\overset{\sim}{R}}_{m}^{\prime\; H}\left( {M - 1} \right)}}},} \\{{a_{2,1} = {{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{c}\left( {M - 1} \right)}\rho_{M,1}R_{c}{{\overset{\sim}{S}}_{c}^{H}(0)}{{\overset{\sim}{R}}_{m}^{\prime\; H}(0)}}},} \\{a_{2,2} = {{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{c}\left( {M - 1} \right)}\rho_{M,M}R_{c}{{\overset{\sim}{S}}_{c}^{H}\left( {M - 1} \right)}{{{\overset{\sim}{R}}_{m}^{\prime\; H}\left( {M - 1} \right)}.}}}\end{matrix} \right\} & (136)\end{matrix}$This is the clutter portion of R_(l) in eqn. (37). Substituting eqn.(135) into eqn. (37) enables using eqn. (48) as the detector 590 todetect a range migrating target in the presence of range migratingclutter 120.

Subsection (c)—Non-Uniform Range Mitigation Detector: Under somegeometries, the range migration of clutter 120 may be significantlydifferent for each range cell 655. Under this condition, the rangemigration matrix is different for each range cell 655. Under thiscondition, the range migration matrix is defined as:

$\begin{matrix}{{{{\overset{\sim}{R}}_{m}\left( {m,i} \right)}_{({k,i})} = {{sinc}\left( \frac{\left( {{\left( {k - 1} \right)T_{s}} - {\Delta\; T_{s}}} \right)}{T_{s}} \right)}},} & (137)\end{matrix}$where i indicates the range cell that the range migration matrix isapplicable and

$\begin{matrix}{{\Delta\; T_{s}} = {\frac{2}{c}{\left( {{R_{GRP}^{i}\left( {{mT}_{i} + {IT}_{s}} \right)} - {R_{GRP}^{i}(0)}} \right).}}} & (138)\end{matrix}$Next the column vector c(t;i) is defined as a vector of length ofP+2(N−1) whose elements are all zero except the i^(th) element. Thiselement is equal to the i^(th) element of c(t). Thus,

$\begin{matrix}{{c(t)} = {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}{{c\left( {t;i} \right)}.}}} & (139)\end{matrix}$

Using these definitions, the receive data Y_(c) due to the clutter 120for pulse m is:

$\begin{matrix}{{y_{c}(m)} = {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}{{{\overset{\sim}{R}}_{m}\left( {m;i} \right)}\;{{\overset{\sim}{S}}_{c}(m)}\;{{c\left( {t,i} \right)}.}}}} & (140)\end{matrix}$where {tilde over (S)}_(c)(m) has been previously defined at eqn. (130).The full vector of received data due to clutter 120 is:

$\begin{matrix}{Y_{t} = {\begin{bmatrix}{y_{c}(0)} \\\vdots \\{y_{c}\left( {M - 1} \right)}\end{bmatrix}.}} & (141)\end{matrix}$Thus the correlation matrix of the clutter 120 in the received data Y:

$\begin{matrix}{R_{Y_{c}} = {{E\left\{ {{\overset{\sim}{Y}}_{c}\mspace{14mu}{\overset{\sim}{Y}}_{c}^{H}} \right\}} = \mspace{149mu}{\begin{bmatrix}{E\left\{ {{y_{c}(0)}{y_{c}^{H}(0)}} \right\}} & \; & {E\left\{ {{y_{c}(0)}{y_{c}^{H}\left( {M - 1} \right)}} \right\}} \\\; & ⋰ & \mspace{11mu} \\{E\left\{ {{y_{c}\left( {M - 1} \right)}{y_{c}^{H}(0)}} \right\}} & \; & {E\left\{ {{y_{c}\left( {M - 1} \right)}{y_{c}^{H}\left( {M - 1} \right)}} \right\}}\end{bmatrix}.}}} & (142)\end{matrix}$Note that:

$\begin{matrix}{{{E\left\{ {{y_{c}(j)}{y_{c}^{H}(k)}} \right\}} = {E\begin{Bmatrix}{\left( {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}{{{\overset{\sim}{R}}_{m}\left( {j,i} \right)}\;{{\overset{\sim}{S}}_{c}(j)}\;{c\left( {{(j)T_{i}};i} \right)}}} \right)\;\ldots} \\{\left( {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}{{{\overset{\sim}{R}}_{m}\left( {k,l} \right)}\;{{\overset{\sim}{S}}_{c}(k)}\;{c\left( {{(k)T_{i}};l} \right)}}} \right)\;}^{H}\end{Bmatrix}}},} & (143)\end{matrix}$which can be factored into:

$\begin{matrix}{{E\left\{ {{y_{c}(j)}{y_{c}^{H}(k)}} \right\}} = {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}{\sum\limits_{j = 1}^{P + {2{({N - 1})}}}{{{\overset{\sim}{R}}_{m}\left( {j,i} \right)}\;{{\overset{\sim}{S}}_{c}(j)}\mspace{11mu} E\left\{ {{c\left( {{(j)T_{i}};i} \right)}{c^{H}\left( {{(k)T_{i}};l} \right)}} \right\}{{\overset{\sim}{S}}_{c}^{H}(k)}\;{{{\overset{\sim}{R}}_{m}^{H}\left( {k,l} \right)}.}}}}} & (144)\end{matrix}$Next one can observe that clutter expectancy:

$\begin{matrix}{{{E\left\{ {{c\left( {{jT}_{i};i} \right)}{c\left( {{kT}_{i};l} \right)}} \right\}} = \lbrack 0\rbrack},} & (145)\end{matrix}$for i≠l. This is due to the previously stated assumption that clutter120 is a zero mean process and is uncorrelated between range cells 655.

On the other hand, if i=l then expectancy concentrates:

$\begin{matrix}{{{E\left\{ {\left( {c\left( {{jT};i} \right)} \right){c^{H}\left( {{jT};i} \right)}} \right\}} = {\rho_{j,k}\begin{bmatrix}0 & \; & \ldots & \; & 0 \\\; & \ddots & \; & \; & \; \\\; & \; & \sigma_{i}^{2} & \; & \; \\\; & \; & \; & \ddots & \; \\0 & \; & \ldots & \; & 0\end{bmatrix}}},} & (146)\end{matrix}$where ρ_(j,k) is defined in eqn. (36). This means that eqn. (146) is amatrix of all zeros except for the i^(th) diagonal value, which is equalto the variance of the clutter 120 at the i^(th) GRP 650 multiplied bythe (j,k)^(th) time correlation coefficient for that range cell. Nextspatial correlation matrix of clutter R_(c)(i) is defined with allelements equal to zero except for the (i,i)^(th) element which is set tothe (i,i)^(th) element of matrix R_(c), namely clutter variances σ_(i)². Note the summation matrix:

$\begin{matrix}{R_{c} = {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}\;{{R_{c}(i)}.}}} & (147)\end{matrix}$With this eqn. (144) can be written as:

$\begin{matrix}{{E\left\{ {{y_{c}(j)}\;{y_{c}^{H}(j)}} \right\}} = {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}\;{{{\overset{\sim}{R}}_{m}\left( {j,i} \right)}\;{{\overset{\sim}{S}}_{c}(j)}\;\rho_{j,k}{R_{c}(i)}\;{{\overset{\sim}{S}}_{c}^{H}(k)}\;{{{\overset{\sim}{R}}_{m}^{H}\left( {k,i} \right)}.}}}} & (148)\end{matrix}$

Applying eqns. (148) to (142) produces:

$\begin{matrix}{R_{Y_{c}} = {\begin{bmatrix}a_{1,1} & \; & a_{1,2} \\\mspace{11mu} & ⋰ & \; \\a_{2,1} & \; & a_{2,2}\end{bmatrix}'}} & (149) \\{where} & \; \\\left. \begin{matrix}{{a_{1,1} = {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}\;{{{\overset{\sim}{R}}_{m}\left( {0,i} \right)}\;{{\overset{\sim}{S}}_{c}(0)}\;\rho_{j,k}{R_{c}(i)}\;{{\overset{\sim}{S}}_{c}^{H}(0)}\;{{\overset{\sim}{R}}_{m}^{H}\left( {0,i} \right)}}}},} \\{{a_{1,2} = {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}\;{{{\overset{\sim}{R}}_{m}\left( {0,i} \right)}\;{{\overset{\sim}{S}}_{c}(0)}\;\rho_{j,k}{R_{c}(i)}\;{{\overset{\sim}{S}}_{c}^{H}\left( {M - 1} \right)}\;{{\overset{\sim}{R}}_{m}^{H}\left( {{M - 1},i} \right)}}}},} \\{{a_{2,1} = {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}\;{{{\overset{\sim}{R}}_{m}\left( {{M - 1},i} \right)}\;{{\overset{\sim}{S}}_{c}\left( {M - 1} \right)}\;\rho_{j,k}{R_{c}(i)}\;{{\overset{\sim}{S}}_{c}^{H}(0)}\;{{\overset{\sim}{R}}_{m}^{H}\left( {k,0} \right)}}}},} \\{{a_{2,2} = {\sum\limits_{i = 1}^{P + {2{({N - 1})}}}\;{{{\overset{\sim}{R}}_{m}\left( {{M - 1},i} \right)}\;{{\overset{\sim}{S}}_{c}\left( {M - 1} \right)}\;\rho_{j,k}{R_{c}(i)}\;{{\overset{\sim}{S}}_{c}^{H}\left( {M - 1} \right)}\;{{\overset{\sim}{R}}_{m}^{H}\left( {{M - 1},i} \right)}}}},}\end{matrix} \right\} & (150)\end{matrix}$This is the clutter portion of Interference R_(l) in eqn. (37).

Subsection (d)—Estimation of Clutter Mean Amplitude: There are a numberof different methods in estimating mean clutter power σ_(i) ². Note thatclutter variance σ_(i) ² must be estimated for each range cell i. Onemethod is to use a clutter model such as the Littoral Clutter Model,which has been validated against a number of data sets and shown togenerally estimate the clutter to noise to within a standard deviationof 10 dB. Another method is to operate the radar 130 to make cluttermeasurements online. This generally called clutter mapping.

Subsection (e)—Estimation of Clutter Spectrum: As with amplitude, thereare a number of different ways to estimate the clutter spectrum or slowtime correlation of clutter (M_(c) and ρ_(j,k)). As with clutteramplitude, the radar 130 can estimate the clutter slow time correlationproperties on-line. A common method to accomplish that is used inweather radar and called pulse pair processing. For radars mounted onfast moving vehicles, the motion of the radar 130 often dominates theclutter spectrum. Under this condition, provides a convenient manner tocalculate the Doppler time correlation properties of ground clutter. Themotion of the antenna relative to the clutter 120 is illustrated in view615.

The motion of the antenna beam 640 across the ground modulates theclutter 120 in addition to the Doppler changes based on angle, see eqn.(118). The clutter correlation function R_(g) ^(i)(τ) from eqn. (20) canbe used. To begin, the transient clutter observed in the receiver 550for a given range resolution cell 655 centered at the i^(th) GRP 650 canbe represented as:

$\begin{matrix}{{{c_{i}(t)} = {\int{{\zeta_{t}(\theta)}{G_{ant}\left( {{\theta - \theta_{P\;\_\;{az}}},{\theta_{el} - \theta_{P\;\_\;{el}}}} \right)}d\;\theta}}},} & (151)\end{matrix}$where c_(i)(t) is the clutter voltage present at the receiver for therange cell of interest, t is slow time, ζ_(t)(θ) is the incrementalclutter response as a function of azimuth angle θ_(az), G_(ant)(θ,φ) isthe two-way (i.e., transmit and receive) antenna voltage gain as afunction of azimuth angle θ_(az) and elevation angle θ_(el), θ_(P_az) isthe azimuth pointing angle of the antenna 535, θ_(P_el) is the elevationpointing angle of the antenna 535. The integration in eqn. (151) sums upthe back-scatter from the ground according to the antenna gain. Thevariable of integration is azimuth (changes of gain over elevation isassumed to be negligible). The angles are referenced at the beginning ofthe CPI.

The clutter voltage response for some time τ after the beginning of theCPI is:

$\begin{matrix}{{{c_{i}\left( {t + \tau} \right)}{\int{{\zeta_{t + \tau}(\theta)}{G_{ant}\left( {{\theta - \theta_{P\;\_\;{az}} - {{\Delta\theta}_{az}(\tau)}},{\theta_{el} - \theta_{P\;\_\;{el}}}} \right)}d\;\theta}}},} & (152)\end{matrix}$where random variable:

$\begin{matrix}{{{\zeta_{t + \tau}(\theta)} = {{\zeta_{t}(\theta)}{\exp\left( {j\; 2\;{{\pi\tau}\left( {{f_{d}\left( {{\theta - {{\Delta\theta}_{az}(\tau)}},\theta_{P\_ el}} \right)} - {f_{d}\left( {\theta_{P\;\_\;{az}},\theta_{P\;\_\;{el}}} \right)}} \right)}} \right)}}},} & (153)\end{matrix}$and Δθ_(az)(τ) is the apparent azimuth change of the GRP 650 over thecourse of the CPI as illustrated in view 615 computed as:

$\begin{matrix}{{{{\Delta\theta}_{az}(\tau)} = {{\theta_{az}(\tau)} - {\theta_{az}(0)}}},} & (154)\end{matrix}$where θ_(az)(r) is defined in eqn. (120) and the Doppler frequency f_(d)is defined in eqn. (118). The reason that the term f_(d)(θ_(P az),θ_(P el)) is subtracted in eqn. (153) is that the mean Doppler isaccounted for in eqn. (131) in the signal convolution matrix {tilde over(S)}_(c) Thus, it must be removed in eqn. (153). The time correlation ofthe clutter for the i^(th) resolution cell associated with the i^(th)GRP 650 is:

$\begin{matrix}{{R_{g}^{i}(\tau)} = {E{\left\{ {{c_{i}(t)}{c_{i}^{*}\left( {t + \tau} \right)}} \right\}.}}} & (155)\end{matrix}$This can be written as correlation:

$\begin{matrix}{{{R_{g\; i}(\tau)} = {E\left\{ {\int{\int{{\zeta_{t}(\alpha)}\;{\zeta_{i + t}^{*}(\beta)}{G_{ant}\left( {{\alpha - \theta_{P\_{az}}},{\theta_{e\; t} - \theta_{P\_{et}}}} \right)}\ldots\mspace{259mu}{G_{ant}^{*}\left( {{\beta - \theta_{P\_{az}} - {{\Delta\theta}_{az}(\tau)}},{\theta_{e\; t} - \theta_{P\_{et}}}} \right)}\; d\;\alpha\; d\;\beta}}} \right\}}},} & (156)\end{matrix}$where α and β are dummy variables of integration.

The order of integration and expectation can be changed to give:

$\begin{matrix}{{{R_{g\; i}(\tau)} = {\int{\int{E\left\{ {{\zeta_{t}(\alpha)}\;{\zeta_{i + t}^{*}(\beta)}} \right\}{G_{ant}\left( {{\alpha - \theta_{P\_{az}}},{\theta_{e\; t} - \theta_{P\_{et}}}} \right)}\ldots}}}}\mspace{259mu}{{G_{ant}^{*}\left( {{\beta - \theta_{P\_{az}} - {{\Delta\theta}_{az}(\tau)}},{\theta_{e\; t} - \theta_{P\_{et}}}} \right)}\; d\;\alpha\; d\;{\beta.}}} & (157)\end{matrix}$Next the expectation can be rewritten as:

$\begin{matrix}{{E\left\{ {{\zeta_{t}(\alpha)}\;{\zeta_{i + t}^{*}(\beta)}} \right\}} = {E{\left\{ {{\zeta_{i}(\alpha)}\;{\zeta_{t}^{*}(\beta)}} \right\} \cdot \exp}\;{\left( {j\;\pi\;{\tau\left( {{f_{d}\left( {{\beta - {\Delta\;{\theta_{az}(\tau)}}},\theta_{P\_{az}},\theta_{P\_{et}}} \right)} - {f_{d}\left( {\theta_{P\_{az}},\theta_{P\_{et}}} \right)}} \right)}} \right).}}} & (158)\end{matrix}$The following assumptions used are applied here, namely:

$\begin{matrix}{{{E\left\{ {\zeta_{i}(\alpha)} \right\}} = 0},{\forall\alpha},} & (159) \\{and} & \; \\{{E\left\{ {{\zeta_{i}(\alpha)}\;{\zeta_{t}^{*}(\beta)}} \right\}} = \left\{ {\begin{matrix}{1,} & {\alpha = \beta} \\{0,} & {\alpha \neq \beta}\end{matrix}\;.} \right.} & (160)\end{matrix}$The assumption in eqn. (159) is arrived at by assuming the phase angleof the clutter voltage is uniformly distributed zero to 2π for everylocation. There are two assumptions in eqn. (160). First the power isunity at all locations. (This is done for convenience because theclutter power is captured in random variables σ_(i)'s.) The secondresult comes from the assumption that the clutter voltage phase isindependent from angle to angle. Applying eqns. (158), (159) and (152)through (157) produces the following for the time correlation function:

$\begin{matrix}{R_{g}^{i} = {\int{{G_{ant}\left( {{\theta - \theta_{P\_{az}}},{\theta_{et} - \theta_{P\_{et}}}} \right)}{{G_{ant}^{*}\left( {{\theta - \theta_{P\_{az}} - \mspace{85mu}{{\Delta\theta}_{az}(\tau)}},{\theta_{et} - \theta_{P\_{et}}}} \right)} \cdot \exp}\;\left( {{- {j2}}\;\pi\;{\tau\left( {{f_{d}\left( {{\theta - {\Delta\;{\theta_{az}(\tau)}}},\theta_{P\_{et}}} \right)} - \mspace{520mu}{f_{d}\left( {\theta_{P\_{az}},\theta_{P\_{et}}} \right)}} \right)}} \right)d\;{\theta.}}}} & (161)\end{matrix}$

Section V—Summary and Results: This study set out to develop signalprocessing that could detect range migrating targets in the presence ofclutter. This has been an open problem. Consequently, eqn. (48) is thesignal processor that was derived for this problem. In support of thiseffort a very general target motion model was developed. (There was norequirement for constant velocity or constant acceleration of the target510.) The signal processor was synthesized by deriving an optimumdetector 590 that jointly processes fast time and slow time data. Thisoptimum detector solves the problem of maximizing the probability ofdetection of a target subject to range and pulse distortion in thepresence of clutter 120.

In order simplify this development, the optimum detector 590 (jointlyprocessing fast time and slow time data) was first developed fornon-range migrating targets. This was summarized in eqn. (34). Thisdetector 590 is a whitening matched filter, and as such it maximizes theSIR. This detector 590 also maximizes the probability of targetdetection for a given probability of false alarm (i.e., being aNeyman-Pearson test). Based on the framework developed, the detector 590for the range migrating targets was derived by modifying the targetmodel to include range migration and pulse distortion. Thus, the signalprocessor that detects range migrating, pulse distorting targets is alsoa whitening matched filter eqn. (48). Because this signal processor is alinear processor, direct performance comparisons with other linearprocessors can be made by comparing the output SIR for each processor.The standard linear processor for radar is the Correlator+MTD. Thisprocessor can be shown to be optimum in AWGN. It is convenient toimplement as it is often implemented with Fast Fourier Transforms (FFT).

To demonstrate this comparison, FIG. 7 provides a graphical view 700that plots optimum detector versus Correlator+MTD. Normalized Dopplerfrequency 710 (in multiples of PRF) denotes the abscissa, while SIR 720(in decibels) denotes the ordinate for a target 510 in AWGN only (i.e.,no clutter). A legend 730 identifies the optimum detector (denoted byopen square symbols a) and correlator+MTD (denoted by diagonal crosssymbols x). These are plotted together along the overlapping lines 740with a point 750 corresponding to Doppler of 0.51 PRF and SIR of 74.771dB. View 700 plots the SIR versus ambiguous Doppler frequency for theoptimum detector 590 and the Correlator+MTD for a target 510 in AWGNonly (i.e., no clutter). In view 700, one can observe that the SIR forboth detectors are the same by lines 740. This is because the optimumdetector reverts to the Correlator+MTD when there is no clutter. Thedata input to the detector are complex data sampled at 80 MHz to satisfythe Nyquist sampling theorem. The waveform consists of thirty samplesduration (N=30) with ten pulses (M=10).

FIG. 8 shows a graphical view 800 of a plot for optimum detector versusCorrelator+MTD for a slow moving target 110 and thus no rangemitigation. Normalized Doppler frequency 810 (multiples of PRF) denotesthe abscissa, while SIR 820 (in decibels) denotes the ordinate, similarto view 700. A legend 830 identifies lines for the optimum detector 840and the Correlator+MTD 850. By contrast, FIG. 9 shows a graphical view900 of a plot for optimum detector versus correation+MTD for a Mach 5target 310, 410 with significant range migration. Normalized Dopplerfrequency 910 (multiples of PRF) denotes the abscissa, while SIR 920 (indecibels) denotes the ordinate, similar to view 700. A legend 930identifies lines for the optimum detector 940 and the correlator+MTD950. The optimum detector profiles 840 and 940 exhibit similar levels at˜72 dB between 0.3 PRF and 0.7 PRF, while the Correlator+MTD profiles850 and 950 reveal significant contrast. For the slow moving target 110,the correlator profile 850 remains level at ˜70 dB from 0.35 PRF to 0.65PRF, while at Mach 5, the correlator profile 950 exhibits a narrow peakat 50 dB at 0.5 PRF.

For the case of a slow moving target 110 in clutter, view 800 plots theSIR versus ambiguous Doppler frequency for the optimum detector and theCorrelator+MTD for a target not subject to range migration. The detector590 for this target as presented in eqn. (34). For slow moving targets(i.e., negligible range migration) there is small gain in performance(˜2 dB) in the range of 0.35 PRF to 0.65 PRF normalized Doppler. Theperformance of the optimum detector, closer to zero Doppler is due tothe fact that it has perfect knowledge of the clutter spectrum asopposed to the Correlator+MTD. Note that this implementation of theCorrelator+MTD in view 800 uses Blackman weighting. The performancegain, of the optimum detector over Correlator+MTD near the center of theDoppler passband, is due to the loss of the Blackman weighting and thegain of the optimum detector over the correlator.

FIG. 9 illustrates a graphical view 900 for the impact of rangemigration in the presence of clutter on the optimum detector developedhere and on the Correlator+MTD. As can be seen in view 900, theCorrelator+MTD suffers a substantial loss in the presence of clutter. Onthe other hand the optimum detector keeps most of its performance forambiguous Doppler of PRF/2 while gaining substantial performance fortarget Dopplers dose to the clutter Doppler. The performance gain attarget Doppler close to the clutter Doppler is due to the detectorfocusing the target 510 while rejecting the clutter 120. The optimumdetector 590 is able to achieve its performance because it jointlyprocesses fast time and slow time data. For an optimum detector 590,fast time and slow time could not be processed separately, but needed tobe processed concurrently.

The present disclosure expands the optimum detector to include rangemigration of the target 510 and the clutter 120. In order to apply thewhitening matched filter in eqns. (34) and (48), the assumption had tobe made that the clutter power and time correlation was known for eachrange resolution cell 655. Advances in clutter modeling enable theclutter parameters to be estimated. In order to evaluate the degradationof the optimum detector 590 in the presence of estimation errors,performance equations were derived to calculate the output SIR under thecondition of errors in clutter parameters, e.g., eqn. (104), whichenables one to calculate the SIR when clutter parameters have beenincorrectly determined. By taking the ratio of eqns. (93) and (85) orelse eqns. (100) and (104), one can calculate the loss induced byclutter parameter estimation errors. First to compare this detectionstrategy, one should consider the case of a non-range migrating target110. In this case, view 800 shows that the optimum detector has nearly a2 dB advantage over the Correlator+MTD. This occurs due to the fact thatpractical implementations of the Correlator+MTD often use weighting tocontrol Doppler side-lobe response and the fact that it is not theoptimum detector in clutter 120. Both of these features cause theCorrelator+MTD to suffer a loss compared to the optimum detector 590.

In order to establish a meaningful assessment of the performance of theoptimum detector with clutter parameter estimation errors, the followingsituation was modeled. First, the actual clutter-to-noise ratio forevery range sample was specified as 30 dB. Next, the error in estimatingthe clutter-to-noise is modeled as a normally distributed error with astandard deviation 10 dB. That is, the error in decibels is normallydistributed with a standard deviation of 10 dB and zero mean. This meansthe linear values of clutter variance σ_(i) ², have a log normal erroradded to it for every range sample. Similarly, the Doppler spectrum isspecified to conform to a text book model of sea clutter, namely, thespectrum is a Gaussian shape with the sigma determined by the sea state.Further, it is modeled that the sea state is estimated with a Gaussianerror. This Gaussian error has a standard deviation of a half sea state.Thus, every element of clutter correlation p_(j,k) is impacted by thesea state estimation error.

FIG. 10 shows a graphical view 1000 of optimum detector performance ofidentical pulses for three distinguished losses for a non-rangemitigating target 110 and with sea state of unity. The comparison isbetween Matched and Mismatched pulses. Clutter-to-noise ratio (CNR) 1010(decibels) denotes the abscissa, while gain 1020 (decibels) denotes theordinate. A legend 1030 identifies mean loss value lines over ten trialsfor average loss 1040, peak loss 1050 and zero bin loss 1060. Thus view1000 quantifies the losses 1040, 1050 and 1060 incurred by optimumdetector for a non-range migrating target when there are clutterparameter errors and plots the results of ten trails for each value ofsea state. The average gain 1040 represents the loss averaged over thewhole ambiguous Doppler space. The peak gain 1050 represents the lossfor targets whose ambiguous Doppler is at the center of the Doppler passband. The zero bin gain 1060 is for targets whose ambiguous Doppler iszero. As can be seen, the loss for all the gains 1040, 1050 and 1060,for all CNRs is 2 dB or less. The conclusion here, under reasonableassumptions of clutter parameter estimation error, that detectionperformance will be on par with a Correlator+MTD. This assertion isjustified because the optimum detector can have a roughly 2 dB advantageover a practical implementation of the Correlator+MTD as in view 800.

FIG. 11 shows a graphical view 1100 of optimum detector performance ofidentical pulses for three distinguished losses for a non-rangemitigating target 110 and CNR of 30 dB. The comparison is betweenMatched detectors that have the correct clutter parameters andMismatched detectors that have estimate of clutter parameters witherrors. Sea state 1110 (logarithmic scale) denotes the abscissa, whilegain 1120 (decibels) denotes the ordinate. A legend 1130 identifies meanloss value lines over ten trials for average loss 1140, peak loss 1150and zero bin loss 1160. Thus view 1100 quantifies the losses 1140, 1150and 1160 incurred by optimum detector for a non-range migrating targetwhen there are clutter parameter errors. To demonstrate that the optimumdetector is robust to clutter estimation errors for different clutteramplitudes, view 1100 plots detector loss versus clutter to noise levelwith the loss of the optimum detector with clutter parameter errors as afunction of the true clutter to noise ratio. For this case the actualsea state is fixed to unity with a half sea state standard deviationerror. As can be seen in view 1100, the mean loss over the trials isless than 2 dB.

FIG. 12 shows a graphical view 1200 of optimum detector performance ofidentical pulses for three distinguished losses for a Mach 5 target andwith 30 dB CNR. Sea state 1210 (logarithmic scale) denotes the abscissa,while gain 1220 (decibels) denotes the ordinate. A legend 1230identifies mean loss value lines over ten trials for average loss 1240,peak loss 1250 and zero bin loss 1260. The performance of the optimumdetector for range migrating, pulse distortion targets under clutterparameter estimation errors is shown in view 1200 with the loss due toestimation error is plotted versus sea state for a nominal target rangerate of Mach 5. The true CNR is 30 dB, with 10 dB standard deviationestimation error. Here, although the losses are somewhat larger than fora slow target 110 in view 1000, they are still very small compared tothe gain in performance of the optimum detector compared to theCorrelator+MTD as shown in view 900.

FIG. 13 shows a graphical view 1300 of optimum detector performance ofidentical pulses for three distinguished losses for a 30 dB CNR and asea state of unity. Target range rate 1310 (Mach) denotes the abscissa,while gain 1320 (decibels) denotes the ordinate. A legend 1330identifies mean loss value lines over ten trials for average loss 1340,peak loss 1350 and zero bin loss 1360, plotting the loss due to clutterparameter error versus nominal target range rate. For this case, theactual sea state is fixed to unity with a half sea state standarddeviation error. In view 1300, one can observe that there can be 5 dB orgreater loss. However, this is balanced against the substantial gainover the optimum detector compared to the Correlator+MTD as shown inview 900. The conclusion again is that this signal processing approachis robust to errors in clutter parameter estimation.

Signal processing to address range migration targets can also detecttargets in clutter when the transmitted signal changes pulse-to-pulse.Other approaches to enable the radar to change signals imposeconstraints on the transmitted signal. They require the range timeside-lobes to be similar to limit the side-lobe clutter modulation thatis not cancelled by the radar's Doppler processor. The signal processingdeveloped here enables changing any waveform modulation frompulse-to-pulse. The detector 590 for this situation was described byeqns. (110) and (111).

FIG. 14 shows a graphical view 1400 of optimum detector performance ofmismatched non-identical pulses versus matched identical pulses forthree distinguished losses for a slow speed target and with sea state ofunity. Mean CNR 1410 (decibels) denotes the abscissa, while gain 1420(decibels) denotes the ordinate. A legend 1430 identifies mean lossvalue lines over ten trials for average loss 1440, peak loss 1450 andzero bin loss 1480, for the performance of the non-identical pulsedetector in the presence of clutter parameter errors. For this case, theactual sea state is fixed to unity with a half sea state standarddeviation error. In view 1400, the loss is less than 2 dB. This loss isagainst an optimum detector with perfect knowledge of the clutterparameters using identical pulses. View 1400 provides a comparison ofmismatched detector using non-identical pulses versus matched detectorusing identical pulses as a function of mean clutter to noise ratio,slow speed target, while constant sea state is unity.

The exemplary detectors 590 developed herein are for situations of rangemigrating clutter 120. This situation arises when the radar 130 is on anaircraft or spacecraft looking down on the ground. The ground clutter480 appears to be moving to the radar 130 and may experience rangemigration. This can be in addition to target range migration and mightbe different amounts of range migration for the target 410 and theclutter 480 depending on the radar and target velocity vectors. Forsituations of uniform clutter range migration, the clutter correlationmatrix is determined by eqn. (135). Substituting eqn. (135) into eqn.(37) enables using eqn. (48) as the detector 590 to detect a rangemigrating target 210, 310, 410 in the presence of range migratingclutter 120.

For the case of clutter range migration that varies with range, eqn.(149) was developed to compute the clutter correlation matrix.Substituting eqn. (149) into eqn. (37) enables using eqn. (48) as thedetector 590 to detect a range migrating target in the presence of rangemigrating clutter. Finally, this disclosure addresses calculating theclutter spectrum for radars in motion. For fast moving radars, theclutter spectrum is dominated by the radar motion. This provides aconvenient way to calculate the clutter time correlation parameters. Thefinal development in this effort enables integrating this technique toairborne and spaceborne radars.

FIG. 15 shows a tabular view 1500 of radar parameters. Note that Table 1in view 1500 lists in separate columns the radar parameters 1510 withvalues 1520 used in the calculation of view 700 and subsequent graphs inSection V. These include operating frequency, bandwidth, number ofpulses, number of chips and PRF. FIGS. 16A, 16B, 16C, 16D, 16E and 16Fshow tabular views 1600 of symbols and their meaning in separatecolumns. The first column 1610 identifies symbol characters. The secondcolumn 1620 provides corresponding definitions. The third column 1630provides an information source for these parameters.

FIGS. 17A, 17B and 17C show a flowchart view 1700 of target detectionprocedures. The process begins with start 1705. The process begins inFIG. 17A with Start 1705 and determines clutter amplitude R_(c) 1710either by modeling or online estimation. This continues to determinetime correlation matrix ρ_(j,k) 1715 by modeling or online estimationfollowed by determine receive noise variance σ_(n) ² 1720. The processcontinues at node A 1725 initiating in FIG. 17B with first decision 1730on whether the clutter moves. If yes, the process calculates cluttersignal convolution matrix {tilde over (S)}_(c)(m) 1735 from cluttermotion from eqns. (130). This continues to calculate the clutter rangemigration matrix {tilde over (R)}_(m)(m) 1740 from clutter motions byeither eqns. (137) or (46). This continues to calculating theinterference correlation matrix R_(l) 1745 by eqns. (135) or (149),either together with eqn. (32). If no, the process calculates theinterference correlation matrix 1750 by eqn. (37).

The process continues at node B 1755 initiating in FIG. 17C withcalculate motion signal convolution matrix {tilde over (S)}_(d)(m) andtarget return phase change u_(i) 1760 from target motion model in eqns.(7), (8) and (23). A second decision 1765 queries whether the target issubject to range migration. If yes, the process calculates the rangemigration matrix {tilde over (R)}_(m)(n) 1770 from the target motion ineqns. (7) and (8), and then proceeds to form the target detector 1775 ineqn. (48) based on threshold η. If no, the process forms the targetdetector 1780 in eqn. (34) based on threshold η. Following operations1775 or 1780 for detector formation for radar 130, the processterminates 1790.

While certain features of the embodiments of the invention have beenillustrated as described herein, many modifications, substitutions,changes and equivalents will now occur to those skilled in the art. Itis, therefore, to be understood that the appended claims are intended tocover all such modifications and changes as fall within the true spiritof the embodiments.

What is claimed is:
 1. A computer-implemented method for detecting atarget amidst clutter by a radar system able to transmit anelectromagnetic signal, receive from a radar antenna first and secondechoes respectively from said target and said clutter, and process saidechoes, said method comprising: determining receive signal convolutionmatrix {tilde over (S)} for the target; determining clutter amplitude byspatial correlation matrix of clutter R_(c); determining timecorrelation matrix ρ_(j,k); determining receive noise variance σ_(n) ²;querying whether the clutter moves as a motion condition when satisfied;calculating interference correlation matrix R_(l) from said receivesignal convolution matrix, said time correlation matrix and said receivenoise variance; calculating doppler signal convolution matrix {tildeover (S)}_(d) and target return phase change u_(i) from said receivesignal convolution matrix phase {tilde over (S)} for target motion;querying whether the target has range migration as a migration conditionfor pulse m when satisfied; and forming a target detector for the radarsystem based on threshold η, wherein for said motion condition furtherincluding prior to said calculating interference correlation matrix:calculating clutter signal convolution matrix {tilde over (S)}_(c)(m)from clutter motion for said pulse m, and calculating clutter rangemigration matrix {tilde over (R)}_(m)(m) for said pulse in from saidclutter motion, and for said migration condition further including priorto said forming said target detector: calculating target range migrationmatrix {tilde over (R)}_(m)(m) from said target motion.
 2. The methodaccording to claim 1, wherein the clutter is modeled by said spatial andtime correlation matrices.
 3. The method according to claim 1, whereinforming a detector employs ${❘{\begin{bmatrix}{u_{0}{{\overset{\sim}{R}}_{m}(0)}{{\overset{\sim}{S}}_{d}(0)}h_{t,{R(0)}}} \\{u_{1}{{\overset{\sim}{R}}_{m}(1)}{{\overset{\sim}{S}}_{d}(1)}h_{t,{R(0)}}} \\ \vdots \\{u_{M - 1}{{\overset{\sim}{R}}_{m}\left( {M - 1} \right)}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}h_{t,{R(0)}}}\end{bmatrix}^{H}R_{I}^{- 1}Y}❘}\begin{matrix}H_{1} \\ > \\ < \\H_{0}\end{matrix}\eta$ for said migration condition and$❘\left. {\begin{bmatrix}{u_{0}{{\overset{\sim}{S}}_{d}(0)}h_{t}} \\ \vdots \\{u_{M - 1}{{\overset{\sim}{S}}_{d}\left( {M - 1} \right)}h_{t}}\end{bmatrix}^{H}R_{I}^{- 1}Y} \middle| {\begin{matrix}H_{1} \\ > \\ < \\H_{0}\end{matrix}\eta} \right.$ where u_(i) is pulse change for each 6 phase,h_(i) is position and impulse response of the target, Y is a stackedvector, and H₀ and H₁ are lower and upper threshold values.
 4. Themethod according to claim 1, wherein said doppler signal convolutionmatrix is determined by: ${{{\overset{˜}{S}}_{d}(m)} = \begin{bmatrix}0 & \ldots & s_{1} & {s_{2}e^{j{\phi({m,1})}}} & \ldots & {s_{N}e^{j{\phi({m,{N - 1}})}}} \\ & & & \vdots & & \\s_{1} & {s_{2}e^{j{\phi({m,1})}}} & \ldots & {s_{N}e^{j{\phi({m,{N - 1}})}}} & \ldots & 0\end{bmatrix}},$ where s₁ are elements of the electromagnetic signalfrom the radar system.
 5. The method according to claim 1, wherein saidtarget return phase change is determined by:${u_{i} = {\exp\left( {j4\pi\frac{R\left( {iT}_{i} \right)}{\lambda}} \right)}},$where λ is the wavelength and T_(i) is the pulse repetition interval.